Singular submodule

Definitions

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

• M is called a singular module if \mathcal{Z}(M) = M,.
• M is called a nonsingular module if \mathcal{Z}(M) = {0},.
• R is called right nonsingular if \mathcal{Z}(R_R) = {0},. 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 \mathcal{Z}(R_R)\subsetneq 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.

Properties

Some general properties of the singular submodule include:

• \mathcal{Z}(M_R) \cdot \mathrm{soc}(R_R) = {0}, where \mathrm{soc}(M_R), denotes the socle of R_R.
• If f is a homomorphism of R-modules from M to N, then f(\mathcal{Z}(M))\subseteq \mathcal{Z}(N),.
• If N is a submodule of M, then \mathcal{Z}(N)=N\cap \mathcal{Z}(M),.
• 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 t(M/t(M))={0},, but this does not necessarily hold for the singular submodule. However, if R is a right nonsingular ring, then \mathcal{Z}(M/\mathcal{Z}(M))={0},.
• 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 \mathcal{Z}(R_R)=J(R),, where J(R) is the Jacobson radical of R.

Examples

Right nonsingular rings are a very broad class, including reduced rings, right (semi)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 (due to R. E. Johnson ) contains several important equivalences. For any ring R, the following are equivalent:

1. R is right nonsingular.
2. The injective hull E(R**R) is a nonsingular right R-module.
3. The endomorphism ring S = \mathrm{End}(E(R_R)), is a semiprimitive ring (that is, J(S)={0},).
4. The maximal right ring of quotients Q_{max}^r(R) 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.

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 Q_{max}^r(R) is a right full linear ring if and only if R has a nonsingular, faithful, uniform module. Moreover, Q_{max}^r(R) is a finite direct product of full linear rings if and only if R has a nonsingular, faithful module with finite uniform dimension.

Textbooks

Primary sources