Singular submodule

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 \mathcal{Z}(M)=\{m\in M \mid \mathrm{ann}(m)\subseteq_e R\}\,. For general rings, \mathcal{Z}(M) 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, tors(M)=\mathcal{Z}(M).

If R is any ring, \mathcal{Z}(R_R) is defined considering R as a right module, and in this case \mathcal{Z}(R_R) is a twosided ideal of R called the right singular ideal of R. Similarly the left handed analogue \mathcal{Z}(_R R) is defined. It is possible for \mathcal{Z}(R_R)\neq\mathcal{Z}(_R R).

Definitions

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

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:

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 (Lam 1999, p. 376)) contains several important equivalences. For any ring R, the following are equivalent:

  1. R is right nonsingular.
  2. The injective hull E(RR) 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. (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 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

This article is issued from Wikipedia - version of the Friday, June 05, 2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.