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Difference between revisions of "Balanced module"

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A module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150801.png" /> such that the natural ring homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150802.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150803.png" /> is regarded as a right module over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150804.png" />, defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150805.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150806.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150807.png" />, is surjective. A module <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150808.png" /> over a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b0150809.png" /> is a generator of the category of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b01508011.png" />-modules if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b01508012.png" /> is balanced as an R-module, projective and finitely generated as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b015/b015080/b01508013.png" />-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.
  
 
====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>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  C. Faith,  "Algebra: rings, modules and categories" , '''1–2''' , Springer  (1973–1976)</TD></TR>
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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
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article