Balanced module

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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.


[1] C. Faith, "Algebra: rings, modules and categories" , 1–2 , Springer (1973–1976)
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Balanced module. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article