Inmathematics, asubalgebra is a subset of analgebra, closed under all its operations, and carrying the induced operations.
"Algebra", when referring to a structure, often means avector space ormodule equipped with an additional bilinear operation. Algebras inuniversal algebra are far more general: they are a common generalisation ofallalgebraic structures. "Subalgebra" can refer to either case.
Asubalgebra of analgebra over a commutative ring or field is avector subspace which is closed under the multiplication of vectors. The restriction of the algebra multiplication makes it an algebra over the same ring or field. This notion also applies to most specializations, where the multiplication must satisfy additional properties, e.g. toassociative algebras or toLie algebras. Only forunital algebras is there a stronger notion, ofunital subalgebra, for which it is also required that the unit of the subalgebra be the unit of the bigger algebra.
The 2×2-matrices over the realsR form a four-dimensional unital algebra M(2,R) in the obvious way. The 2×2-matrices for which all entries are zero, except for the first one on the diagonal, form a subalgebra. It is also unital, but it is not a unital subalgebra.
Theidentity element of M(2,R) is theidentity matrix I , so the unital subalgebras contain the line ofdiagonal matrices {x I :x inR}. For two-dimensional subalgebras, consider
Whenp = 0, then E isnilpotent and the subalgebra {x I +y E :x, y inR } is a copy of thedual number plane. Whenp is negative, takeq = 1/√−p, so that (q E)2 = − I, and subalgebra {x I +y (qE) :x,y inR } is a copy of thecomplex plane. Finally, whenp is positive, takeq = 1/√p, so that (qE)2 = I, and subalgebra {x I +y (qE) :x,y inR } is a copy of the plane ofsplit-complex numbers. By thelaw of trichotomy, these are the only planar subalgebras of M(2,R).
L. E. Dickson noted in 1914, the "Equivalence ofcomplex quaternion and complex matric algebras", meaning M(2,C), the 2x2 complex matrices.[1] But he notes also, "the real quaternion and real matric sub-algebras are not [isomorphic]." The difference is evident as there are the threeisomorphism classs of planar subalgebras of M(2,R), while real quaternions have only one isomorphism class of planar subalgebras as they are all isomorphic toC.
Inuniversal algebra, asubalgebra of analgebraA is asubsetS ofA that also has the structure of an algebra of the same type when the algebraic operations are restricted toS. If the axioms of a kind ofalgebraic structure is described byequational laws, as is typically the case in universal algebra, then the only thing that needs to be checked is thatS isclosed under the operations.
Some authors consider algebras withpartial functions. There are various ways of defining subalgebras for these. Another generalization of algebras is to allow relations. These more general algebras are usually calledstructures, and they are studied inmodel theory and intheoretical computer science. For structures with relations there are notions of weak and of inducedsubstructures.
For example, the standard signature forgroups in universal algebra is(•,−1, 1). (Inversion and unit are needed to get the right notions of homomorphism and so that the group laws can be expressed as equations.) Therefore, asubgroup of a groupG is a subsetS ofG such that: