Difference between revisions of "Balanced module"
From Encyclopedia of Mathematics
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| − | A module | + | A module $M$ such that the natural ring homomorphism $\phi : R \rightarrow \mathrm{End}_{\mathrm{End}_R M} M$, where $M$ is regarded as a right module over $\mathrm{End}_R M$, defined by $\phi(r)(m) = mr$ for any $r \in R$ and $m \in M$, is surjective. A module $P$ over a ring $R$ is a generator of the category of $R$-modules if and only if $P$ is balanced as an $R$-module, projective and finitely generated as an $\mathrm{End}_R P$-module. |
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules and categories" , '''1–2''' , Springer (1973–1976)</TD></TR></table> | + | <table> |
| + | <TR><TD valign="top">[1]</TD> <TD valign="top"> C. Faith, "Algebra: rings, modules and categories" , '''1–2''' , Springer (1973–1976)</TD></TR> | ||
| + | </table> | ||
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| + | {{TEX|done}} | ||
Latest revision as of 20:23, 10 October 2017
A module $M$ such that the natural ring homomorphism $\phi : R \rightarrow \mathrm{End}_{\mathrm{End}_R M} M$, where $M$ is regarded as a right module over $\mathrm{End}_R M$, defined by $\phi(r)(m) = mr$ for any $r \in R$ and $m \in M$, is surjective. A module $P$ over a ring $R$ is a generator of the category of $R$-modules if and only if $P$ is balanced as an $R$-module, projective and finitely generated as an $\mathrm{End}_R P$-module.
References
| [1] | C. Faith, "Algebra: rings, modules and categories" , 1–2 , Springer (1973–1976) |
How to Cite This Entry:
Balanced module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Balanced_module&oldid=18292
Balanced module. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Balanced_module&oldid=18292
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article