# Essential submodule

From Encyclopedia of Mathematics

Revision as of 20:13, 30 October 2016 by Richard Pinch (talk | contribs) (Start article: Essential submodule)

*of a module $M$*

A submodule $E$ of $M$ is essential it has a non-trival intersection with every non-trivial submodule of $M$: that is, $E \cap L = 0$ implies $L = 0$.

Dually, a submodule $S$ is **superfluous** if it is not a summand of $M$: that is, $S + L = M$ implies $L = M$.

See also: Essential subgroup.

#### References

- F.W. Anderson, K.R. Fuller, "Rings and Categories of Modules" Graduate Texts in Mathematics
**13**Springer (2012) ISBN 1468499130

**How to Cite This Entry:**

Essential submodule.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Essential_submodule&oldid=39549