# Semi-chain module

From Encyclopedia of Mathematics

A module which can be decomposed as a direct sum of chain submodules (see Chain module). All left $R$-modules are semi-chain modules if and only if $R$ is a generalized uniserial ring.

#### Comments

A chain module is also called a uniserial module; correspondingly, a semi-chain module can also be called a semi-uniserial module. In [a1] the terms $\Sigma$-uniserial and $\sigma$-uniserial are used for a direct sum of uniserial modules (respectively, a finite direct sum of uniserial modules).

#### References

[a1] | C. Faith, "Algebra" , II. Ring theory , Springer (1976) pp. Chapt. 25 |

**How to Cite This Entry:**

Semi-chain module.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Semi-chain_module&oldid=32109

This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article