In the branches of abstract algebra known as ring theory and module theory, each right (resp.
left) R-module M has a singular submodule consisting of elements whose annihilators are essential right (resp.
left) ideals in R. In set notation it is usually denoted as
is a good generalization of the torsion submodule tors(M) which is most often defined for domains.
In the case that R is a commutative domain,
is defined considering R as a right module, and in this case
is a two-sided ideal of R called the right singular ideal of R. The left handed analogue
In the following, M is an R-module: In rings with unity it is always the case that
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.
Johnson's Theorem (due to R. E. Johnson (Lam 1999, p. 376)) contains several important equivalences.
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.
is a right full linear ring if and only if R has a nonsingular, faithful, uniform module.
is a finite direct product of full linear rings if and only if R has a nonsingular, faithful module with finite uniform dimension.