Namespaces
Variants
Actions

Difference between revisions of "Semi-perfect ring"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
(TeX)
 
Line 1: Line 1:
−
A [[Ring|ring]] over which every finitely-generated left (or every finitely-generated right) module has a projective covering. A ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084270/s0842701.png" /> with [[Jacobson radical|Jacobson radical]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084270/s0842702.png" /> is a semi-perfect ring if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084270/s0842703.png" /> is semi-local and if every idempotent of the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084270/s0842704.png" /> has an idempotent pre-image in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084270/s0842705.png" />. The first condition can be replaced by the requirement of classical semi-simplicity of the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084270/s0842706.png" /> (cf. [[Classical semi-simple ring|Classical semi-simple ring]]), and the second by the possibility of  "lifting"  modular direct decompositions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084270/s0842707.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084270/s0842708.png" />. A semi-perfect ring may also be characterized by the condition that every module admits a direct decomposition with respect to which the maximal direct summands are complemented. A ring of matrices over a semi-perfect ring is a semi-perfect ring.
+
{{TEX|done}}
 +
A [[Ring|ring]] over which every finitely-generated left (or every finitely-generated right) module has a projective covering. A ring $R$ with [[Jacobson radical|Jacobson radical]] $J$ is a semi-perfect ring if and only if $R$ is semi-local and if every idempotent of the quotient ring $R/J$ has an idempotent pre-image in $R$. The first condition can be replaced by the requirement of classical semi-simplicity of the quotient ring $R/J$ (cf. [[Classical semi-simple ring|Classical semi-simple ring]]), and the second by the possibility of  "lifting"  modular direct decompositions from $R/J$ to $R$. A semi-perfect ring may also be characterized by the condition that every module admits a direct decomposition with respect to which the maximal direct summands are complemented. A ring of matrices over a semi-perfect ring is a semi-perfect ring.
  
 
See also [[Perfect ring|Perfect ring]], and the references to that article.
 
See also [[Perfect ring|Perfect ring]], and the references to that article.

Latest revision as of 16:20, 4 October 2014

A ring over which every finitely-generated left (or every finitely-generated right) module has a projective covering. A ring $R$ with Jacobson radical $J$ is a semi-perfect ring if and only if $R$ is semi-local and if every idempotent of the quotient ring $R/J$ has an idempotent pre-image in $R$. The first condition can be replaced by the requirement of classical semi-simplicity of the quotient ring $R/J$ (cf. Classical semi-simple ring), and the second by the possibility of "lifting" modular direct decompositions from $R/J$ to $R$. A semi-perfect ring may also be characterized by the condition that every module admits a direct decomposition with respect to which the maximal direct summands are complemented. A ring of matrices over a semi-perfect ring is a semi-perfect ring.

See also Perfect ring, and the references to that article.


Comments

Cf. also Projective covering.

References

[a1] L.H. Rowen, "Ring theory" , 1 , Acad. Press (1988) pp. 217
How to Cite This Entry:
Semi-perfect ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Semi-perfect_ring&oldid=33494
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
Retrieved from "https://encyclopediaofmath.org/index.php?title=Semi-perfect_ring&oldid=33494"