nLab uniform module

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

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

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

References

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

category: algebra