Semi-chain module

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