Some general properties of the singular submodule include:
- where denotes the socle of .
- If f is a homomorphism of R-modules from M to N, then .
- If N is a submodule of M, then .
- The properties "singular" and "nonsingular" are Morita invariant properties.
- The singular ideals of a ring contain central nilpotent elements of the ring. Consequently, the singular ideal of a commutative ring contains the nilradical of the ring.
- A general property of the torsion submodule is that , but this does not necessarily hold for the singular submodule. However, if R is a right nonsingular ring, then .
- If N is an essential submodule of M (both right modules) then M/N is singular. If M is a free module, or if R is right nonsingular, then the converse is true.
- A semisimple module is nonsingular if and only if it is a projective module.
- If R is a right self-injective ring, then , where J(R) is the Jacobson radical of R.
Johnson's Theorem (due to R. E. Johnson (Lam 1999, p. 376)) contains several important equivalences. For any ring R, the following are equivalent:
- R is right nonsingular.
- The injective hull E(RR) is a nonsingular right R-module.
- The endomorphism ring is a semiprimitive ring (that is, ).
- The maximal right ring of quotients is von Neumann regular.
Right nonsingularity has a strong interaction with right self injective rings as well.
Theorem: If R is a right self injective ring, then the following conditions on R 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: If R is a ring, then is a right full linear ring if and only if R has a nonsingular, faithful, uniform module. Moreover, is a finite direct product of full linear rings if and only if R has a nonsingular, faithful module with finite uniform 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