uniform module

Given a (possibly noncommutative) unital ring RR (left or right) nonzero RR-module MM is uniform if the intersection of any two nonzero submodules of MM is nonzero, or, equivalently, such that every nonzero submodule of MM is essential in MM.

A nonzero module MM is uniform iff its injective envelope E(M)E(M) is indecomposable. In particular, if MM is itself injective, then it is uniform iff it is indecomposable.

An arbitrary RR-module MM has finite rank if and only if it has an essential submodule which is a finite direct sum of uniform submodules.

  • K. R. Goodearl, R. B. Warfield Jr. An introduction to noncommutative Noetherian rings, London Math. Soc. Student Texts 61, 2nd ed. 2004
category: algebra

Last revised on February 26, 2014 at 06:59:05. See the history of this page for a list of all contributions to it.