In the branches ofabstract algebra known asring theory andmodule theory, each right (resp. left)R-moduleM has asingular submodule consisting of elements whoseannihilators areessential right (resp. left)ideals inR. In set notation it is usually denoted as${\displaystyle \mathcal{Z}(M)=\{m\in M\mid \mathrm{a}\mathrm{n}\mathrm{n}(m){\subseteq }_{e}R\}}$. For generalrings,${\displaystyle \mathcal{Z}(M)}$ is a good generalization of thetorsion submodule tors(M) which is most often defined fordomains. In the case thatR is acommutative domain,${\displaystyle \mathrm{tors}(M)=\mathcal{Z}(M)}$.

IfR is any ring,${\displaystyle \mathcal{Z}(R_R)}$ is defined consideringR as a right module, and in this case${\displaystyle \mathcal{Z}(R_R)}$ is a two-sided ideal ofR called theright singular ideal ofR. The left handed analogue${\displaystyle \mathcal{Z}{(}_{R}R)}$ is defined similarly. It is possible for${\displaystyle \mathcal{Z}(R_R)\ne \mathcal{Z}{(}_{R}R)}$.

Here are several definitions used when studying singular submodules and singular ideals. In the following,M is anR-module:

• M is called asingular module if${\displaystyle \mathcal{Z}(M)=M}$.
• M is called anonsingular module if${\displaystyle \mathcal{Z}(M)=\{0\}}$.
• R is calledright nonsingular if${\displaystyle \mathcal{Z}(R_R)=\{0\}}$. Aleft nonsingular ring is defined similarly, using the left singular ideal, and it is entirely possible for a ring to be right-but-not-left nonsingular.

In rings with unity it is always the case that${\displaystyle \mathcal{Z}(R_R)⊊R}$, and so "right singular ring" is not usually defined the same way as singular modules are. Some authors have used "singular ring" to mean "has a nonzero singular ideal", however this usage is not consistent with the usage of the adjectives for modules.

Some general properties of the singular submodule include:

• ${\displaystyle \mathcal{Z}(M_R)\cdot \mathrm{s}\mathrm{o}\mathrm{c}(R_R)=\{0\}}$ where${\displaystyle \mathrm{s}\mathrm{o}\mathrm{c}(M_R)}$ denotes thesocle of${\displaystyle R_R}$.
• Iff is ahomomorphism ofR-modules fromM toN, then${\displaystyle f(\mathcal{Z}(M))\subseteq \mathcal{Z}(N)}$.
• IfN is asubmodule ofM, then${\displaystyle \mathcal{Z}(N)=N\cap \mathcal{Z}(M)}$.
• The properties "singular" and "nonsingular" areMorita invariant properties.
• The singular ideals of a ring contain centralnilpotent elements of the ring. Consequently, the singular ideal of a commutative ring contains thenilradical of the ring.
• A general property of the torsion submodule is that${\displaystyle t(M/t(M))=\{0\}}$, but this does not necessarily hold for the singular submodule. However, ifR is a right nonsingular ring, then${\displaystyle \mathcal{Z}(M/\mathcal{Z}(M))=\{0\}}$.
• IfN is an essential submodule ofM (both right modules) thenM/N is singular. IfM is afree module, or ifR is right nonsingular, then the converse is true.
• Asemisimple module is nonsingularif and only if it is aprojective module.
• IfR is a rightself-injective ring, then${\displaystyle \mathcal{Z}(R_R)=J(R)}$, where J(R) is theJacobson radical ofR.

Right nonsingular rings are a very broad class, includingreduced rings, right(semi)hereditary rings,von Neumann regular rings,domains,semisimple rings,Baer rings and rightRickart rings.

For commutative rings, being nonsingular is equivalent to being a reduced ring.

Johnson's Theorem (due to R. E. Johnson (Lam 1999, p. 376)) contains several important equivalences. For any ringR, the following are equivalent:

1. R is right nonsingular.
2. Theinjective hull E(R_R) is a nonsingular rightR-module.
3. Theendomorphism ring${\displaystyle S=\mathrm{E}\mathrm{n}\mathrm{d}(E(R_R))}$ is asemiprimitive ring (that is,${\displaystyle J(S)=\{0\}}$).
4. Themaximal right ring of quotients${\displaystyle Q_{max}^r(R)}$ is von Neumann regular.

Right nonsingularity has astrong interaction with right self injective rings as well.

Theorem: IfR is a right self injective ring, then the following conditions onR are equivalent: right nonsingular, von Neumann regular, right semihereditary, right Rickart, Baer, semiprimitive. (Lam 1999, p. 262)

The paper (Zelmanowitz 1983) used nonsingular modules to characterize the class of rings whose maximal right ring of quotients have a certain structure.

Theorem: IfR is a ring, then${\displaystyle Q_{max}^r(R)}$ is a rightfull linear ring if and only ifR has a nonsingular,faithful,uniform module. Moreover,${\displaystyle Q_{max}^r(R)}$ is a finite direct product of full linear rings if and only ifR has a nonsingular, faithful module with finiteuniform dimension.

• Goodearl, K. R. (1976),Ring theory: Nonsingular rings and modules, Pure and Applied Mathematics, No. 33, New York: Marcel Dekker Inc., pp. viii+206,MR 0429962
• Lam, Tsit-Yuen (1999),Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York:Springer-Verlag,doi:10.1007/978-1-4612-0525-8,ISBN 978-0-387-98428-5,MR 1653294

• Zelmanowitz, J. M. (1983), "The structure of rings with faithful nonsingular modules",Trans. Amer. Math. Soc.,278 (1):347–359,doi:10.2307/1999320,ISSN 0002-9947,MR 0697079

• Module theory
• Ring theory