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Essential submodule

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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://www.encyclopediaofmath.org/index.php?title=Essential_submodule&oldid=39549