Difference between revisions of "Semi-chain module"
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| − | A [[Module|module]] which can be decomposed as a direct sum of chain submodules (see [[Chain module|Chain module]]). All left | + | {{TEX|done}} |
| + | A [[Module|module]] which can be decomposed as a direct sum of chain submodules (see [[Chain module|Chain module]]). All left $R$-modules are semi-chain modules if and only if $R$ is a generalized [[Uniserial ring|uniserial ring]]. | ||
====Comments==== | ====Comments==== | ||
| − | A chain module is also called a uniserial module; correspondingly, a semi-chain module can also be called a semi-uniserial module. In [[#References|[a1]]] the terms | + | A chain module is also called a uniserial module; correspondingly, a semi-chain module can also be called a semi-uniserial module. In [[#References|[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==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Faith, "Algebra" , '''II. Ring theory''' , Springer (1976) pp. Chapt. 25</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Faith, "Algebra" , '''II. Ring theory''' , Springer (1976) pp. Chapt. 25</TD></TR></table> | ||
Latest revision as of 16:40, 1 May 2014
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=17890
Semi-chain module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-chain_module&oldid=17890
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article