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(2019-04-20)  
(Assuming some familiarity with the modern concept of abstract rings.)

Abstract rings  were introduced in 1871 by Richard Dedekind (1831-1916)  under the name of Ordnung.  Dedekind also pioneered the related notions of module  and ideal (then limited to prime ideals) in the context of what we would now recognize as the integers of a number field.

In 1892, David Hilbert (1862-1942) coined the modern name  (Zahlring,  then Ring)  which first appeared in print in 1897. The German word Ring  and the English word ring share with circle  two connotations which may have inspired Hilbert: One denotes a group of people,  the other evokescircularity and looping back  (as in the rings of nonzerocharacteristics  which Hilbert was then concerned with). The latter etymology is dominant,  as shown by the fact that it's solelyresponsible for the French translation  (anneau)  which lacks the first connotation.

In 1914, Abraham Fraenkel (1891-1965) introduced an axiomatic definition which was far more restrictive than the current one,  as it required the existence of a multiplicative identity and postulatedthat every noninvertible element should be a divisor of zero (which rules out ordinary integers).

In 1917,  the Japanese mathematicianMasazo Sono (1886-1969)  dropped those requirementsbut retained commutativity,  quoting compatibility with the field axioms  published in 1903 by L. Eugene Dickson (1874-1954).

In 1921, Emmy Noether (1882-1935)  did the same thingindependently.  A few years later,  she finally dropped the requirement of commutativity.

Rings with a multiplicative neutral element are the most important type of rings and many authors  will only study those.  They're variously known as "rings with 1", "rings with unity", "rings with identity", "unit rings", "unitary rings"  or  "unital rings"  (which is the only term I'll use).

For any given  (abelian)  additive group  G, the ring of square zero  G(0) is the ring in which the product of two elements is always zero. Bourbaki  had to call it a pseudo-ring of square zero  (pseudo-anneau de carré nul) because,  in their work,  the bare term ring  was reserved for unital rings. This trivial case plays a key role in the enumeration  of finite rings.

Reputable authors will routinely start a specialized discussion with a fair warning like: "Throughout this paper,  the symbol R  will denote a finite associativering with multiplicative identity."  That's the way to go...

(2006-02-15)  
Addition, subtraction and multiplication are defined,  division needn't be.

(A, +,. )  is a ring  when addition  (+)  and multiplication  (.) are well-defined  internal operationsover the set A  with the following properties:

• (A,+)  is a commutativegroup  whose neutral element is called  0  (zero).
• Multiplication is an associative (not necessarily commutative)  internal operation which is distributive  over addition.  That's to say:
  x  y  z      x.(y + z)   =   x.y + x.z       and       (x + y).z   =   x.z + y.z

Multiplicative  notationsallow the omission of the dot  symbol  (.).

Optional properties of a ring can be indicated by specific qualifiers:

• Unital Ring :   There's a multiplicative neutral element:  1.x = x.1 = x
• Commutative Ring  :  x  y   x.y = y.x
• Integral Ring :   The product of two nonzero elements is nonzero.
• Division Ring :   Any nonzero element has a multiplicative inverse.

There seems to be universal agreement to define an integral domain  asa  commutative integral ring. The current trend in Ring Theory (echoed by Wikipedia) is to call a domain  any integral ring and specify that the term integral domain  only applies to commutative ones. I still advise against the term domain  outside idioms like integral domain.

A field  is normally defined as a commutative division ring  (a division ring where multiplication is commutative)  unless otherwise specified. I regard as synonymous the locutions noncommutative division ring and skew field  (as well as the semi-acceptable oxymoron  of noncommutative field). Some authors allow  commutativity in a skew field, in part to translate what the French call a field  (corps) which is a division ring,  commutative or not.

Lesser Rings :

For the record,  some algebraic structures have been defined which are endowed with an additionand a multiplication distributive  over it, but without some of the other requirements imposed on rings.  Examples:

A semiring  is built on an additive monoid instead of an additive group. This is to say that a semiring  contains a zero element (neutral for addition)  but subtraction need not be well-defined. In a semiring,  0  is postulated  to be absorbent (x,   0.x = x.0 = 0) which is a theorem  in a ring. The prototypical example of a semiring is  (,+,×)  the set of natural integers,  endowed with usual addition and multiplication.

  =   { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, ... }

Also common are the tropical semirings (e.g., the max-plus algebra  where  - is included as neutral additive element with the new operations defined by xy = max(x,y)   and  xy = x+y). Likewise in idempotent semirings xx = x  holds for any  x. Those may be called semirings of characteristic one (the only ring  of characteristic  one isthe trivial field).

A near-ring  is a ring-like structure where the additive groupis not necessarily abelian  (i.e., addition needn't be commutative). Multiplication must still be associative but distributivity may not hold on both sides. Thus,  in a right near-ring  multiplication is onlyrequired to right-distribute  over addition,  which is to say:

x  y  z      (x + y).z   =   x.z + y.z

For example,  over an additive group  G  (abelian or not) the functions  from  G  to  G form a right near-ring  if the addition of functions is defined pointwise  and multiplication is understood as the composition  of functions:

xG        (f +g ) (x)   =  f (x)  + g (x)             (f .g ) (x)   =  f (g (x) )

Those definitions imply right-distributivity  (f +g ) .h  = f.h +g.h   since:

xG        (f +g ) (h (x))   =  f (h (x))  + g (h (x))    

(2006-02-15)   zero-divisors :
In some rings,  the product of two nonzero  elements can be zero.

In a ring, by definition,  an element d  is said to be a divisor  of a given element a  when there is a nonzero  element  x  such that:

d x   =  a       or       xd   =  a

In particular  (with a = 0 )  a divisor of zero is an element whose product into some nonzero element is equal to zero. As usual,  a proper divisor can't be equal to the dividend  itself. So,  the zero element itself isn't a proper  divisor of zero. Zero is sometimes called a trivial  divisor of zero.

• d  is a left divisor of zero  when there's a nonzero  v  such that  dv = 0.
• d  is a right divisor of zero  when there's a nonzero  u  such that  ud = 0.
• d  is a two-sided divisor of zero  when both exist  (whether  u = v  or not).
• d  is a regular element  if neither  proposition is true.

Many authors,  including myself,  use the locution zero-divisor  to denote a proper divisor of zero (more awkwardly dubbed nontrivial divisor of zero  or nonzero divisor of zero). I strongly recommend the hyphenation to best indicate that this locution must be taken as a whole.

Likewise,  left zero-divisors or right zero-divisors are always understood to be nonzero... There are no zero-divisors in integral rings (including integral domains, division rings, fields  and skew fields).

Two elements  x  and  y  are said to be  (mutually) orthogonal  when:

x y   =   0        and        y x   =   0

If some  n^th  power of an element is zero,that element is said to be nilpotent. Clearly, a nonzero nilpotent element is azero-divisor.

x ⁿ   =   0

Thesimplest example  isthe residue 2  in the ring  /4 (i.e.,  the ring formed by the four possible residues of integersmodulo 4) where:

2 ²   =   0

One example of a ring with zero-divisors but no nilpotent  elements is the ring of polyadic integers for any radix g  that's not  a prime-power.

Regular element = cancellable element

Those are the names given to elements which are not  divisisorof zero to a specified side.  Equivalently, muliplication to one sideby an element which is regular to that side is injective. For example,  if the element  r  is right-regular, matching right-factors of  r  can be cancelled from an equation:

x r   =   y r      iff       x   =   y

No side is specified when a regular element is regular to both sides.

(2019-05-11)  

In a finite unital  ring,  every non-invertibleelement is a divisor of zero. To put it bluntly,  every element divides either zero or unity.

 For any element a  of the  finite unital ring A  {0},  ifthe map  which sends  x  to a x isinjective, then it's also surjective  (since A  is finite)  so there's anelement  v  whose image is  1,  meaning a v = 1.

On the other hand,  if that map isn't injective,  then two different elements  x  and  y  have the same image and x-y = v  verifies a v = 0 with  v  0.

Likewise,  considering the map which sends  x  to  x a,  we seethat there's a nonzero element  u  such that  u a is either  0  or  1.

We can't have  u a = 1  and a v = 0  with a nonzero  v (or else we'd have  v = u a v = 0)  or vice-versa. Therefore,  either a  is a unit (with the same inverse on both sides  u = u a v = v) or it's a two-sided divisor of zero.  

The units of a unital ring A  form a multiplicative groupedenoted A*  or  U(A)  (sometimes  E(A) as the German name for unit  is Einheit). That's just a special case of the group M*  formed byall the invertible elements in a multiplicativemonoid M, so the same notation  can be used.

(2006-02-15)  
An ideal is a multiplicatively absorbent  additive subgroup.

An  ideal  is an additive subgroup that containsa product whenever it contains a factor. (Such a thing is called multiplicatively absorbent,  or absorbent  for short.)

A subring  is a ring contained in another one  (using the same operations). With the traditional definition of a ring adopted here  (where we don't require a ring to be unital) a subring is simply a nonempty subset closed under subtraction  and multiplication. That much is clearly true for an ideal.  Thus,  ideals can also be defined as absorbent subrings.

For a left-ideal I,the product ax  is inI whenever x is:  aA,aI  I
For a right-ideal I,the product  xa  is inI whenever x is:  aA,Ia  I
Unless otherwise specified,an ideal  is both  a right-ideal and a left-ideal.

Every ring is a (two-sided) ideal of itself,  called the unit ideal. Any other ideal  (one-sided or two-sided)  is said to be proper.

All rings with more than one element have at least one proper ideal, namely the single-element subring  {0}  which is called the zero ideal. The ring  {0}  doesn't have any proper ideals. The unit ideal  and the zero ideal are both called trivial  (whether or not they are the same).

The sum and theintersection of two same-sided idealsare ideals on that side.

If I  is a left-ideal and J  is a right-ideal, then I.J  is a double-sided ideal,  but neither J.I  nor IJ  are necessarily one-sided ideals.

A  one-sided or two-sided ideal is called maximal when it's a proper  ideal not contained in any other proper ideal of the same kind.  Thus,  the whole ring is never a maximal ideal of itself.  {0}  is maximal only if the ring is nonzero anddoesn't have any other proper ideals of the relevant kind.

A ring A  is said to be a filial ring  when any ideal of an idealof A  is also an ideal of A  itself. Using Wielandt's symbols  that's to say:

J ⊲I ⊲A    J ⊲A

So,  ring A  is filial  when all its 2-accessible  subrings are ideals,  defining a subring S as n-accessible  when a sequence A_i exists which verifies the following  (such an S  is precisely  n-accessible ifnot k-accessible for k<n):

S   =  A₀  ⊲ A₁  ⊲   ...   ⊲ A_n   =  A

The ideal generated  (to one side)  by some set  X  is well-defined as theintersection of all ideals  (to that side)  containing  X, since any intersection of ideals of a given kind is an ideal of the same kind.

An ideal generated  by a single element is called a principal ideal. One example of such a right-ideal is the set   A  of allright-multiples of the element    in the ringA (e.g.,  2  is the ideal of all even integers).

A ring,  like, whose ideals are all  principal is called a principal ring. Such a ring is called a principal integral domain (abbreviated PID)  if it has noproper divisors of zero (i.e., the product of two nonzero elements is never zero). Note that Bourbaki  requiresa principal ring  to be a PID.

In a field,  there are only two ideals, namely  {0}  and the whole field.  They are both principal (respectively generated by the elements  0  and  1)  so a fieldis a  PID. (:  If an ideal of a field contains a nonzero elementit also contains  the product of that element by its inverse,  which is  1.) Every skew field  is a PID too.

Ideals were introduced in 1871 by RichardDedekind (1831-1916)  as he investigated what are now called completely prime ideals,  namely ideals which don't contain aproduct unless  they contain at least one factor (e.g.,  amongintegers,the multiples of aprime number  have that property). In commutative rings,  those are just prime ideals, namely ideals which don't contain a product of two ideals unless they contain at least one of them. A  completely prime ideal  is always a prime ideal, but the converse may not be true in the noncommutative case.  That distinction was introducedin 1928 by Wolfgang Krull (1899-1971).

Theradical  Rad(I)  of an ideal I is the set of all ring elements which have one of their powers in I. The radical of an ideal is an ideal.  If an ideal isthe radical of another it's called a radical ideal. Every prime ideal  is a radical ideal. Modulo a radical ideal, there are no nilpotent residues.

In particular,  the ideal  Rad({0}),  called the nilradical of A,  is the set of all nilpotent elements of A. It's the intersection of all prime ideals  of A.

The Jacobson radical  J(A)  of a ring A  is the intersection of all the maximal ideals  of A. Since all maximal ideals  are prime,  the nilradical  is contained in the Jacobson radical.

(2006-02-15)  
The ring A/I  which consists of all residue classes modulo I.

Modulo anidealI of a ringA,the residue-class (or just residue ) [x] of an element  x  of A is the set of all elements  y  of A for which  x-y  is in I.

The set of all residues moduloI is denotedA/I. It's a ring, variously calledquotient ring,factor ring,residue-class ring or simplyresidue ring.

For example,  /4  is the ring formed bythe four residue classesmodulo 4, whose additionand multiplication tables are shown at right. (Note that "2" is a nilpotent divisor of zero.)

0 1 2 3 0 0 1 2 3 1 1 2 3 0 2 2 3 0 1 3 3 0 1 2

0 1 2 3 0 0 0 0 0 1 0 1 2 3 2 0 2 0 2 3 0 3 2 1

The ring   / p = ( / p, +, ×)  is a field  if and only if  p is prime.

In particular,  the Boolean field   /2  has just two elements;  0  and  1 (called bits  nowadays). It's used below  to construct a nontrivial ring, which provides concrete examples of many abstract concepts.

(2019-04-19)  
Function f  from one ring to another,  which respects the ring operators:

f (x+y)   =  f (x)  + f (y)           f (x y)   =  f (x)  f (y)

If such an homomorphism  is bijective, it's called an isomorphism. An isomorphism from one ring to itself is called an automorphism.

Anti-homomorphisms  reverse multiplication:  f (x y)   =  f (y)  f (x)

If f  is an homomorphism from ring A  to ring B and J  is a subring  of B,  then I = f ^-1 (J)  is a subring. It's an ideal  of A if J  is an ideal of B. I  is called the contraction  of J.

:  In particular, f ^-1 ({0_B})  is an ideal, called the kernel  of f :

ker f     =    f ^-1 ({0})    =    { xA  | f (x) = 0 }

(2006-06-13)  
It's either  0  or the least  p>0  for which all sums of  p  like terms vanish.

In a unital  ring A, we may call "1" the neutral element for multiplication and name  the elementsof the following sequence after ordinary integers:

1 1,  2 1+1,  3 1+1+1,  4 1+1+1+1,  5 1+1+1+1+1,  ...

If all the elements in this sequence are nonzero, the ring is said to havezero characteristic. Otherwise, the vanishing indices  are multiples of the least of them, which is called the characteristic  of the ring, denoted  char(A).

The only ring of characteristic  1  is the trivialfield  (where  1 = 0).

The characteristic of a nontrivial unital ring withoutzero-divisorsis either zero or a prime number.  (: any integer  (1+1+...)  corresponding to a prime divisor  of a composite  characteristic would be a zero-divisor.)

In particular, the characteristic of any nontrivialfield  (or skew-field) is either  0  or a prime number.

The characteristic of a non-unital ring is defined as the least positive integer p  such that a sum of  p  identical terms always vanishes (if there's no such  p,  then the ring is said to have zero characteristic).

The characteristic of a ring depends only  on itsadditive group  (e.g.,  if that additive group is the cyclic group C_n , then the characteristic is  (n), the reduced totient  of  n).

A finite  ring can't have zero characteristic. The characteristic of a finite ring always divides its number of elements. (cf.structure of abelian groups).

Frobenius Map  (1880):

If thecharacteristic  p of a  commutative ring  is a prime number,  we have:

( x y ) ^p   =   x ^p y ^p        and        (x + y) ^p   =   x^p + y^p

The former relation is due to commutativity. The latter relation comes from Newton's binomial formula,  with the added remarkthat thebinomial coefficient  C(p,k) is divisible by  p,  if  p  is prime,  unless  k  is  0  or  p.

Thus,  the map defined by   F(x)  =  x^p  respects  both addition and multiplication. This ring automorphism, is called the Frobenius map,  in honor of F. Georg Frobenius  (1849-1917) who discovered the relevance of such things to algebraic number theory,  in 1880.

Theautomorphism group of theGalois field GF(pⁿ )  is a cyclic group  oforder  n, generated  by the above Frobenius map.

(2019-01-27)   Unital Noncommutative ring).
It can be represented by  8 triangular 2 by 2 boolean  matrices.

We may construct it as a vector space  over the integers modulo 2  by choosing any basis of three independent matrices. This gives an additive group  (of characteristic  2) isomorphic to the  8  integers from  0  to  7  endowed with bitwise  addition  (just like the additive group of the Galois field  GF(8) which see for the explicit addition table).

We'll use the ring isomorphism  defined by the followingcorrespondence between powers of two and lower-triangular binary 2×2 matrices (we also give equivalent upper-triangular matrices highlighted in yellow to make explicit the isomorphism between the two types of triangular matrices). 

Upper Triangular	1  0  0  1	0  1  0  0	1  0  0  0
Lower Triangular	1  0  0  1	0  0  1  0	0  0  0  1
Binary Name	1	2	4

Choices for 1  and 2  were engineeredto give the names  0,1,2 and 3 to the four elements whose unique properties make them invariant under any ring automorphism. Those form a commutative subring  whose multiplication table  is thus made identical to the multiplication table of  /4.

Four possible matrices  can then be assigned to 4. I picked one arbitrarily,  as I'm still looking for a good reason to distinguish one as canonical.

			0  0  0  0	1  0  0  1	0  1  0  0	1  1  0  1	1  0  0  0	0  0  0  1	1  1  0  0	0  1  0  1
			0  0  0  0	1  0  0  1	0  0  1  0	1  0  1  1	0  0  0  1	1  0  0  0	0  0  1  1	1  0  1  0
			0	1	2	3	4	5	6	7
0  0  0  0	0  0  0  0	0	0	0	0	0	0	0	0	0
1  0  0  1	1  0  0  1	1	0	1	2	3	4	5	6	7
0  1  0  0	0  0  1  0	2	0	2	0	2	0	2	0	2
1  1  0  1	1  0  1  1	3	0	3	2	1	4	7	6	5
1  0  0  0	0  0  0  1	4	0	4	2	6	4	0	6	2
0  0  0  1	1  0  0  0	5	0	5	0	5	0	5	0	5
1  1  0  0	0  0  1  1	6	0	6	2	4	4	2	6	0
0  1  0  1	1  0  1  0	7	0	7	0	7	0	7	0	7

Let's use this example as an opportunity to review the basic concepts:

• Element  1  is neutral for multiplication:  1 x  =  x 1  =  x   for any x.
• Element  2  is nilpotent.  {0,2}  is a ring of square zero.
• As expected,  all elements are either divisors of zero  or units  (1 and 3).
• Besides  2  and  3,  all elements are idempotent.
• 4  and  5  are mutually orthogonal.  So are  6  and  7.
• The two trivial ideals  are  {0}  and the unit ideal  {0,1,2,3,4,5,6,7}.
• The zero ideal  {0}  is the only proper  trivial ideal  (as always).
• The only nontrivial  two-sided  ideals are  {0,2},  {0,2,4,6}  and  {0,2,5,7}.
• Besides the two-sided ideals,  the only left-ideals are  {0,4}  and  {0,6}.
• Besides the two-sided ideals,  the only right-ideals are  {0,5}  and  {0,7}.
• The only maximal  ideals are  {0,2,4,6}  and  {0,2,5,7}.
• The subrings  which are not single-sided ideals are  {0,1}  and  {0,1,2,3}.
• {0,2,4,5,6,7}  is multiplicatively absorbent but it isn't an additive subgroup.
• There are  2 ring automorphisms  and  2 anti-automorphisms,  namely:

The automorphisms and anti-automorphisms of the  SUN  ring :

	0	1	2	3	4	5	6	7
Identity	0	1	2	3	4	5	6	7
Automorphism	0	1	2	3	6	7	4	5
Anti-automorphisms	0	1	2	3	5	4	7	6
	0	1	2	3	7	6	5	4

As both anti-automorphisms are involutions,  they can be used to give  SUN the structure of an involutive ring,  in two different ways. Using the Cayley-Dickson construction with either choice yields an algebraic stucture of 16 elements which is not  a ring (the composite structure isn't multiplicatively associative since basic multiplication isn't commutative).

If f (x)  is the number of  1's  in the diagonal  of  [the matrix for]  x,  then:

f (x y)   ≤   min (f (x), f (y) )

	1	i	j
1	1	i	j
i	i	0	0
j	j	j	j

More generally,  for any commutative  unital ring A, we may use the above pattern to define  SUN(A)  as a 3-dimensional module  over A, where multiplication is determined by the multiplication table at right,  for abasis  (1,i,j)  of the module.

For short,  SUN_q = SUN( GF(q) ) when  q  is a prime-power.
Furthermore,  SUN = SUN₂  denotes the ring discussed above.

For a prime  p,  SUN_p  is the only unital noncommutative ring of order  p³.

---

Alternately,  we could have chosen to define the  SUN  ring as generated by two distinct  elements  u  and  v  obeying the following relations:

u²  =  u        v²  =  v        u v  =  v        v u  =  u

With the above numbering,  that's only true when  {u,v} = {4,6}.

We could also have used the relations below,  characterizing  {u,v} = {5,7} :

u²  =  u        v²  =  v        u v  =  u        v u  =  v

(2019-02-14)  
An idempotent  element splits a ring into adirect sum of  4  components.

An idempotent  element of a ring A  is a solution of the equation:

x ²   =   x

Every ring has at least one idempotent element  (namely  0)  and every unital ring  with more than one element has another trivial one  (namely  1).

If a unital ring has other idempotent elements  (said to be nontrivial) then it has at least one zero-divisor  because, in a unital ring,  the above equation reduces to the following zero productof two factors,  neither of which is zero when  x  is neither 0 nor 1:

x (1-x)   =   0

Left-multiplication or right-multiplication by an idempotent  is analogous toa geometrical projection  (projecting a projected image doesn't change it). This geometrical analogy suggests that a ring can be expressed as a direct sum of the imagesof two such complementary projections,  as shown next.

Peirce Decompositions  (Benjamin Peirce, 1870) :

If e  is an idempotent  of the unital ring A (i.e.,  e² = e)  then A can be split into a direct sum  in two different ways (usingMinkowski notations):

• The left Peirce decomposition :  A   =   eA    (1-e)A
• The right Peirce decomposition :  A   =  A e   A (1-e)

In the noncommutative case,  those two are distinct and we may apply them successively toobtain the full  (two-sided) Peirce decomposition  of A  into four components, best presented formally as a matrix:

A
			eA e (1-e)A e		eA (1-e) (1-e)A (1-e)

The right-hand-side denotes a set of  2×2  matrices whose coefficients arein four subsets of A  specified by Minkowski operations. Addition and multiplication of such matrices reduce to component operations in A  according to the usual rules. (It's a simple exercise to show that this set of matrices is stable under additionand multiplication.)

The decomposition can be trivial  (e.g.,  with  e = 1). The reader may want to work out the example   e = 4 with numbered elements of the SUN ring, which recovers a representation in terms of triangular boolean matrices:

SUN2   =  A

4A 4 5A 4

4A 5 5A 5

=

{0,4} {0}

{0,2} {0,5}

More generally,  what we did for   e₁ = e   and  e₂ = 1-e   can be done for any set of  n pairwise orthogonal idempotents adding up to unity:

e_i  e_j   = _ij e_j             e₁  +  e₂  +  ... +  e_n   =   1

A
			e1A e1 e2A e1 ... enA e1		e1A e2 e2A e2 ... enA e2		... ...   ...		e1A en e2A en ... enA en

A more intricate decomposition for Jordan algebras  was introduced in 1947 by A.A. Albert (1905-1972).

(2019-01-25)  
How many different rings with  n  elements are there?

To count the number of rings of order  n (up to isomorphism)  we first factor  n  into  m  primes :

n   =   q₁^k₁q₂^k₂ ...q_m^k_m

A ring is unital  (resp. commutative) if and only if all  its components are.  So, the same enumeration method applies either to unrestricted rings or tothose which are required to be unital and/or commutative. The ensuing four possible types of enumerations are tabulated below in separate columns.

The only possible additive group for a ring of prime  order  p is the cyclic group C_p . As the characteristic  of a finite ring must divide its order, it's either 1 or p.  This yields only two possible rings.  Both are commutative:

• The ring of square zero  C_p(0)   is not unital:  x,  y,   x y  =  0
• The Galois field   GF(p)  = F_p  = /p  is a unital ring,  of course.

There are  11  different rings of order  p², for any prime p.  Of those, 9  are commutative and  4  are unital  (all the unital ones are commutative):

• 3  such rings are decomposable:   (C_p(0))²,  C_p(0) ×F_p  and  (F_p)²
  Only the last one is unital.  All three are commutative.
• 5  other rings over the additive group   (C_p)²   are indecomposable. These,  too,  are algebras  of dimension  2  over the finite field F_p :
  • The two unital ones are commutative:  F_p²   and  GR(p,2)
  • One is commutative but not unital.
  • The other two are neither commutative nor unital.
• The  3  indecomposable rings over the group C_p²  are commutative.
  Only  /p²  is unital.

The enumeration for order  p³  is far more delicate. Preliminary results were published in 1947  by Robert F. Ballieu,  a leading professor at Université catholique de Louvain and a former foreign student at ENS  (Ulm, 1936). A first attempt at a complete enumeration was published in 1973 by Gilmer  &  Mott. It was corrected by Antipkin  &  Elizarov  in1982: 20 such rings are decomposable and either 32  (for p=2)  or  3p+30  (for p≥3)  are not.

The entries in bold  were first obtained by Christof Nöbauer  before 2002.

Because the corresponding enumerations are multiplicative functions (entirely determined by their values at prime-powers)  the above tableis sufficient to enumerate rings of all orders below  64 (except  32  for rings not required to be unital). This yields the following four tables:

By comparing the last two tables,  we see that the unique smallest unital noncommutative ring has 8  elements;  it's the SUN ring studied above. Next up,  there are  13  distinct noncommutative unital rings of order  16. Noncommutative unital rings exist only for orders divisible by a cube :

8, 16, 24, 27, 32, 40, 48, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 120, 125, 128,135, 136, 144, 152, 160, 162, 168, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243,248, 250, 256, 264, 270, 272, 280, 288, 296, 297, 304...(A046099)

Indeed,  if   n  =  k p³ , then one example of a noncommutative unital ring of order  n is given by the direct product  SUN_p × /k

All unital rings of cubefree order  n  (A004709)  are commutativebecause the above enumerations show that there are as just as many commutative unital rings of order  n  as there are unital rings  of the same order. (  That much is true when  n  is a prime or the square of a prime.)

Likewise all rings of squarefree order  (A005117)  are commutative. Noncommutative rings exist only for orders divisible by a square :

4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50,52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99,100, 104, 108, 112, 116, 117, 120, 121, 124, 125, 126, 128, 132, 135, 136, 140,144, 147, 148...(A013929)

(2019-01-25)  
They're called quasi-simple  by Bourbaki.

Ideals  are to rings what divisors are to integers or what normal subgroups are to groups, inasmuch as they allow a standard decomposition when nontrivial examples exist. Otherwise,  we indicate irreducibility to anything simpler by calling those things simple rings, primes  or simple groups.

{0}  isn't considered a simple ring,  for the same reason the number  1  isn't called a prime or  {e}  isn't considered a simple group:  If this convention wasn't made, fundamental theorems would be tougher to state  (and use).

(2019-02-12)  
Two ideals I  and J  are called coprime  when I + J = A.

The term comaximal  is more rarely used for the same concept, which generalizes coprime integers  in a waywhich allows the following generalization of the Chinese remainder theorem.

Generalized Chinese Remainder Theorem

(2019-02-13)  
Rings in which the sum of two principal ideals  is again principal.

(2019-01-18)  
They are to prime ideals  what prime-powers are to prime numbers.

A primary decomposition  is an expression of an ideal as an intersection of finitely many primary ideals. It's analogous to the factorization of a positive integer into a product of prime-powers (fundamental theorem of arithmetic).

The fundamental theorem about primary decompositions is the Lasker-Noether theorem,  which was the main motivation forthe definition of Noetherian rings  (introduced below).

By definition,  an ideal I  is said to be primary when,  for any product x y  in it,  either  x  or some power of y  is also in I.

(2006-04-27)  
A well-defined internal operation among sequences in a ring.

The Cauchy product of two sequences (a₀ ,a₁ ,a₂ , ...)  and (b₀ ,b₁ , ...) of elements from a ring A  is the sequence  (c₀ ,c₁ ,c₂ , ...)   where:

cn   =		aibn-i

Namely: c₀ =a₀b₀ ,  c₁ =a₀b₁ +a₁b₀ ,  c₂ =a₀b₂ +a₁b₁ +a₂b₀ ,  etc.

The set A, of the sequences whose terms are elements of the ring A has the structure of a ring  (dubbed formal power series  overA) if endowed with direct addition  (the n-th term of a sum being the sum of the n-th terms of thetwoaddends)  and the Cauchy multiplication  defined above.

The set A^{( )} consists of those sequences which have only finitely many nonzero terms. It forms asubring of the above ring,better known as the  [univariate] polynomials over A, denoted A[x] and discussednext.

(2006-04-06)  
It's endowed withcomponent-wise addition and Cauchy multiplication.

A finite  sequence of elements of a ring A (or, equivalently, a sequence with finitely many nonzero elements) is called a polynomial  over A. The set of all such polynomials is a ring (often denoted A[x] where  x  is a "dummy variable") which is a subring of theaforementioned ring of"formal power series",underdirect addition andCauchy multiplication.

Each term of the sequence defining a polynomial is called a coefficient. The degree  of a polynomial is the highest rankof its nonzero coefficients  (ranks start at zero). The null polynomial ("zero") has no nonzero coefficients and its degreeis defined to be   (negative infinity). Thus,  for polynomials over an integral ring (a ring withoutdivisors of zero) the degree of a product is always the sum of the degrees of the factors.

If the ring A  is commutative,  the polynomials so definedmatch very well the intuition we may have acquired by using  X as the symbol for a so-called unknown variable  and formingnew expressions by addition and multiplication using  X  andsome constants  taken from the ring A.

In the noncommutative case,  this intuition fails unless we imagine that X  commutes with every element of A. This strange thing is what it means  to state that the productof two polynomials is equal to their Cauchy product.

Formal Polynomials  vs.  Polynomial Functions :

To a polynomial (a₀ ,a₁ ...a_n ) of degree  n,  we associate afunction f :

f (x)   =		ai xi

However, that function and the polynomial which defines itare two different things entirely... For example, over thefinite field  GF(q), the distinct  polynomials  x  and  x^q correspond to the same  function.

Whenever the distinction between a polynomial and itsassociated function must be stressed, the former may be called a formal polynomial.

Over a noncommutative ring, the concept of polynomials doesn't break down, butthe above association of a polynomial with a function is dubious at best, in particular because the value of a product of polynomials at some point neednot be the product of their values at that point.

Polynomial Functions over a Noncommutative Ring

Arguably,  some polynomial functions are not even associated with aformal polynomial in the above sense. Indeed,  we could consider functions which are obtained fromvariables and constants by some fixed recipe involving a finite numberof additions and multiplications,  without assuming anytype of commutativity among constants or variables. This topic is under-investigated in the literature.

Twisted Polynomial Ring

It's the set of univariate polynomials with coefficients appearing formally at the leftof the powers of the variable  X  if multiplication is defined usinga nontrivial  automorphism   in the ring of coefficients  (e.g.,  theFrobenius map  in a commutative ring of coefficientsof nonzero characteristic)  with the following postulated  formal identity:

Xb   =   (b) X

In other words,  the  n-th coefficient of the twisted product  is:

cn   =		ai  (bn-i )

Since    isn't trivial, multiplication of twisted polynomials isn't commutative even when the ring of coefficients is (it stays Noetherian). 

(2022-10-15)  
Bombieri product of two homogeneous multivariate polynomials.

(2019-01-18)  
Rings which don't contain any infinite ascending chains  of ideals.

Four concepts can be defined,  which use the concept of an ascending chain, defined below.  They coincide for commutative  rings:

• Weakly Noetherian :   No infinite ascending chain of ideals  exists.
• Left-Noetherian :   No infinite ascending chain of left-ideals
• Right-Noetherian :   No infinite ascending chain of right-ideals
• Strongly Noetherian :  Both left-Noetherian  and right-Noetherian.

An ascending chain  of sets is defined as a sequence of sets  I_n  whereevery set is contained in its successor,  which is to say:

I₀    I₁    I₂    I₃     ...  I_n    I_n+1     ...

The non-commutative case is under-investigated becausethe Lasker-Noether theorem  (discussed in the next section) doesn't apply to it. Emmy Noether  herself gave a non-commutative example ofa right-Noetherian  ring where some ideals don't have a primary decomposition.

Equivalently,  Noetherian rings can be defined as:

• Rings in which every set of ideals has a maximal  element. (with respect to set-inclusion only;  it needn't be a proper maximal ideal).
• Finitely-generated rings.  (In the commutative case,  at least.)

Trivially,  all finite  rings are Noetherian. Not all Bézout rings  are.

Another example of a (commutative) Noetherian ring is the ring of integers . Indeed,  all its ideals are of the form n (the principal ideal  generated by  n). Thus,  any ascending chain of ideals corresponds to a decreasing sequence of divisors, which is necessarily finite,  so the  ACC  holds.

More generally,  any principal ring  (or PID)  is Noetherian. In particular,  every field  is a Noetherian ring.

Hilbert's Basis Theorem  (Hilbert, 1888) :

Univariate polynomials  over a Noetherian ring  form a Noetherian ring.
Corollary  (byinduction) :  So do multivariate  polynomials.

The theorem is true also in the noncommutative case. Let's give a proof which doesn't depend on commutativity:

Let A  be a Noetherian ring  (using one of the four possible definitions  presented above). A[X]  is the ring formed by all univariate polynomials  with coefficients in A. Consider any ascending chain  of ideals in A[X] :

I₀    I₁    I₂    I₃     ...  I_n    I_n+1     ...

To prove the theorem,  we must establish that this chain cannot be infinite (i.e., it must stabilize after some large enough index).

Well,  we may define an ideal  J_ij  in A as the set of all the coefficients of  X^j  in thepolynomials of  I_i  with degree  i  or less. It's not difficult to prove that every  J  is an ideal  (one-sided ordouble-sided)  in the same sense as every  I  is. (That exercise is left to the reader.)

Furthermore,  it's clear that,  if  i ≤ i'  and  j ≤ j'   then J_i,j     J_i',j'

In spite of its modern terminology,  the above proof is probably quite close tothe nonconstructive proof originally proposed by Hilbert, who was motivated by the abstract generalization of results in the theory of invariants which had been painstakingly derived by Paul Gordan (1837-1912) ofClebsch-Gordan_coefficients fame. Legend has it that Gordan paid a strange compliment to this beautiful proof:

This is not mathematics,  this is theology.

Gröbner bases   (Bruno Buchberger,  1965)

(2019-01-21)  
Rings which don't contain any infinite descending chains  of ideals.

For rings,  surprisingly,  the Artinian descending chain condition  (DCC) implies theNoetherian ascending chain condition  (ACC) but turns out to be strictly stronger. The following statement clarifies the situation and makes the independent study of Artinian rings all but superfluous:

An Artinian rings is a Noetherian ring
where every non-invertible element is nilpotent.

Artinian rings may also be characterized as Noetherian rings whose Krull dimensions  are not positive.

The importance of Artinian rings stems from the fact that it's the natural concept to use inthe Artin-Wedderburn classification theorem.

An Artinian ring is a finite product of local rings (Chriestenson, Tuley, Wane).

(2019-01-18)     (Lasker 1905,  Noether 1921)
Every commutative Noetherian ring  is a Lasker ring  (Laskerian ring).

By definition,  a Lasker ring  is a commutative ring in which any ideal  has a primary decomposition (which is to say that it's the intersection of finitely many primary ideals). In other words,  a generalization of the fundamental theorem of arithmetic holds in such rings.  In 1905, Emanuel Lasker  considered only polynomials  and convergent power series over a field. He proved that the rings they form have that property.  In 1921, Emmy Noether  generalized the theorem to all commutative Noetherian rings.

Lasker didn't point out the unicity of the decomposition,  which was established in 1915 by Francis Macaulay (1862-1937).

The algorithm commonly used to perform actual primary decompositions is due to Grete Hermann (1901-1984). It was part of the doctoral work she completed under Emmy Noether at Göttingen  in 1926.

(2019-01-22)  
In them,  two elements always have a greatest common divisor  (GCD).

As the name implies,  a GCD domain  is a commutative ring.

(2019-01-22)  
GCD domains  with no infinite ascending chains  of principal ideals.

Bourbaki  called such a ring a factorial ring, which is now the only term used in French  (namely, anneau factoriel ).

In a ring,  two related concept  (irreducible and prime) can be defined which coincide only in the case of a factorial rings  (UFD). In such a structure,  a straight counterpart of the fundamental theorem of arithmetic  hold, which justifies the term of unique factorization domain.

• An irreducible  element  q  is an element which only has trivial  factorizations  (i.e.,  it can only be written as a productof two factors if at least one of those is a unit).
• A prime  element  p  is an element which can't dividea product of two factors unless is divides one of them.

The ring  [5] (pronounced  adjoined root  5)  isn't a  UFD because it contains irreducible  elements which aren't prime. For example,  the integer  2  divides the following productwithout dividing either factor:

4   =   (-1 + 5) (1 + 5)

In 1847,  the failure to distinguish between those two concepts famously led the French physicist Gabriel Lamé(1795-1870;X1814) to propose an erroneous proof of Fermat's last theorem. A few years earlier  (1843) Eduard Kummer (1810-1895) had already identified this trap,  which would become one of the key motivations forthe development of ring theory.

(2019-04-23)  
Integral domain where every nontrivial ideal is a product of prime ideals.

An integral domain is a Dedekind domain if, and only if, every nonzerofractional ideal is invertible.

(2006-04-05)  
The modulo-q polynomialsmoduloan irreducible polynomial modulo p.

Let  q  be a power of a prime  p.  Let  f be some monic polynomialmodulo q, of degree  r,  which is irreducible modulo p (i.e., f (x) mod p  never vanishes). The Galois ring  of characteristic  q  and rank  r  isdefined as:

GR(q,r)   =  (/q)[x] /f (x)

Finite noncommutative rings can be described as algebra over Galois rings.

(2014-12-06)  
In those, everyprime ideal isan intersection of maximal ideals.

In particular, an Hilbert subring ring is the intersection of all maximal ideals containing it.

(2019-05-03)  
Rings in which every idempotent element is central.

Abelian rings need not be commutative but they have some properties of commutative rings.

(2018-01-25)     (Krull, 1938)
Local algebra  is the study of commutative local ringsand theirmodules.

A local ring  is a ring which has only one maximal ideal.

This concept was introduced in 1938 by Wolfgang Krull (1899-1971) who called them Stellenringe  in German. The English term local ring  was coined by Oscar Zariski (1899-1986).

(2019-02-03)  

(2020-04-30)  
When the topology  makes addition and multiplication continuous.

When a ring A  is a topological space, its cartesian square A×A  is considered endowed with the product topology  (of Tychonoff). The addition and multiplication are functions from A×A to A.  When both are continuous, A  is said to be a topological ring.

(2020-05-09)  
What a ring  is called when endowed with an involution which is both  an additive automorphism and a multiplicative anti-automorphism.

By definition,  an involution  is just a bijection  equal to its own inverse. In the context of  *-rings,  the image by the aforementioned involutionof an element  x  is denoted  x*  (pronounced x-star) and is called the adjoint  of  x  (or its conjugate). It's now customary to call the  involution thefunction which maps an element to its adjoint (I still call it conjugation  at times).

That's to say that the following axioms are postulated:

• X**   =   X     (i.e.,  "conjugation" is an involution).
• (X + Y)*   =   X* + Y*     (additive homomorphism).
• (X Y)*   =   Y* X*     (multiplicative antihomomorphism).

(2020-04-30)  

(2025-02-28)     and Morita duality.
In topological rings.

Two rings are Morita-equivalent when their categories of modules are additively equivalent.