In the branches of abstract algebra known as ring theory and module theory, each right R module M has a singular submoduleconsisting of elements whose annihilators are essential right ideals in R. In set notation it is usually denoted as. For general rings, is a good generalization of the torsion submodule tors which is most often defined for domains. In the case that R is a commutative domain,. If R is any ring, is defined considering R as a right module, and in this case is a twosided ideal of R called the right singular ideal of R. Similarly the left handed analogue is defined. It is possible for.
Definitions
Here are several definitions used when studying singular submodule and singular ideals. In the following, M is an R module:
M is called a singular module if.
M is called a nonsingular module if.
R is called right nonsingular if. A left 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, 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.
Properties
Some general properties of the singular submodule include:
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 then M/N is singular. If M is a free module, or if R is right nonsingular, then the converse is true.
Right nonsingular rings are a very broad class, including reduced rings, right hereditary rings, von Neumann regular rings, domains, semisimple rings, Baer rings and right Rickart rings. For commutative rings, being nonsingular is equivalent to being a reduced ring.
Important theorems
Johnson's Theorem contains several important equivalences. For any ring R, the following are equivalent:
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. The paper 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.
