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An associative ring such that every left [[Module|module]] over it has a projective covering (cf. [[Associative rings and algebras|Associative rings and algebras]]). Right perfect rings are defined similarly. A left perfect ring need not be right perfect.
 
An associative ring such that every left [[Module|module]] over it has a projective covering (cf. [[Associative rings and algebras|Associative rings and algebras]]). Right perfect rings are defined similarly. A left perfect ring need not be right perfect.
  
The following properties of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721001.png" /> are equivalent: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721002.png" /> is a left perfect ring; 2) every set of pairwise orthogonal idempotents of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721003.png" /> is finite, and every non-zero right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721004.png" />-module has a non-zero [[Socle|socle]]; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721005.png" /> satisfies the minimum condition for principal right ideals; 4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721006.png" /> satisfies the minimum condition for finitely-generated right ideals; 5) every right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721007.png" />-module satisfies the minimum condition for finitely-generated submodules; 6) the [[Jacobson radical|Jacobson radical]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721008.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p0721009.png" /> vanishes on the right (that is, for any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210010.png" /> of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210011.png" /> there is an integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210012.png" /> such that the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210013.png" />), and the quotient ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210014.png" /> is Artinian semi-simple; 7) every flat left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210015.png" />-module is projective; 8) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210016.png" /> contains idempotents <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210017.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210019.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210020.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210021.png" /> is a [[Local ring|local ring]] for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210022.png" />; 9) every left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210023.png" />-module satisfies the maximum condition for cyclic submodules; 10) for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210024.png" /> every left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210025.png" />-module satisfies the maximum condition for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210026.png" />-generated submodules; and 11) every projective left <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210027.png" />-module has a decomposition with respect to which every direct summand has a complement (see [[Krull–Remak–Schmidt theorem|Krull–Remak–Schmidt theorem]]).
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The following properties of a ring $R$ are equivalent: 1) $R$ is a left perfect ring; 2) every set of pairwise orthogonal idempotents of $R$ is finite, and every non-zero right $R$-module has a non-zero [[Socle|socle]]; 3) $R$ satisfies the minimum condition for principal right ideals; 4) $R$ satisfies the minimum condition for finitely-generated right ideals; 5) every right $R$-module satisfies the minimum condition for finitely-generated submodules; 6) the [[Jacobson radical|Jacobson radical]] $J$ of $R$ vanishes on the right (that is, for any sequence $a_1,a_2,\dots,$ of elements of $J$ there is an integer $n$ such that the product $a_1\dots a_n=0$), and the quotient ring $R/J$ is Artinian semi-simple; 7) every flat left $R$-module is projective; 8) $R$ contains idempotents $e_1,\dots,e_n$ such that $\sum_{i=1}^ne_i=1$, $e_ie_j=0$ for $i\neq j$, and $e_iRe_i$ is a [[Local ring|local ring]] for every $i$; 9) every left $R$-module satisfies the maximum condition for cyclic submodules; 10) for every $n$ every left $R$-module satisfies the maximum condition for $n$-generated submodules; and 11) every projective left $R$-module has a decomposition with respect to which every direct summand has a complement (see [[Krull–Remak–Schmidt theorem|Krull–Remak–Schmidt theorem]]).
  
A ring of matrices over a perfect ring is perfect. Idempotent ideals of a perfect ring are generated by idempotents that are central modulo the radical. A group ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210028.png" /> (see [[Group algebra|Group algebra]]) is perfect if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210029.png" /> is a perfect ring and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210030.png" /> is a finite group. The endomorphism ring of an Abelian group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210031.png" /> is perfect only when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210032.png" /> is the direct sum of a finite group and finitely many copies of the additive group of rational numbers. Local perfect rings are characterized by the fact that any linearly independent subset of any free left module over it can be extended to a base. The following properties are also equivalent: a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210033.png" /> is a perfect ring and all its quotient rings are self-injective (cf. [[Self-injective ring|Self-injective ring]]); b) every quotient ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210034.png" /> is quasi-Frobenius (cf. [[Quasi-Frobenius ring|Quasi-Frobenius ring]]); c) every quotient ring of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210035.png" /> has a cogenerator; and d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p072/p072100/p07210036.png" /> is uniserial (cf. [[Uniserial ring|Uniserial ring]]).
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A ring of matrices over a perfect ring is perfect. Idempotent ideals of a perfect ring are generated by idempotents that are central modulo the radical. A group ring $RG$ (see [[Group algebra|Group algebra]]) is perfect if and only if $R$ is a perfect ring and $G$ is a finite group. The endomorphism ring of an Abelian group $A$ is perfect only when $A$ is the direct sum of a finite group and finitely many copies of the additive group of rational numbers. Local perfect rings are characterized by the fact that any linearly independent subset of any free left module over it can be extended to a base. The following properties are also equivalent: a) $R$ is a perfect ring and all its quotient rings are self-injective (cf. [[Self-injective ring|Self-injective ring]]); b) every quotient ring of $R$ is quasi-Frobenius (cf. [[Quasi-Frobenius ring|Quasi-Frobenius ring]]); c) every quotient ring of $R$ has a cogenerator; and d) $R$ is uniserial (cf. [[Uniserial ring|Uniserial ring]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Kasch,  "Modules and rings" , Acad. Press  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''1–2''' , Springer  (1973–1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.T. Markov,  A.V. Mikhalev,  L.A. Skornyakov,  A.A. Tuganbaev,  "Modules"  ''J. Soviet Math.'' , '''23''' :  6  (10983)  pp. 2642–2706  ''Itogi Nauk. Algebra. Topol. Geom.'' , '''19'''  (1981)  pp. 31–134</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  F. Kasch,  "Modules and rings" , Acad. Press  (1982)  (Translated from German)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  C. Faith,  "Algebra" , '''1–2''' , Springer  (1973–1979)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.T. Markov,  A.V. Mikhalev,  L.A. Skornyakov,  A.A. Tuganbaev,  "Modules"  ''J. Soviet Math.'' , '''23''' :  6  (10983)  pp. 2642–2706  ''Itogi Nauk. Algebra. Topol. Geom.'' , '''19'''  (1981)  pp. 31–134</TD></TR></table>

Revision as of 12:26, 28 August 2014

left

An associative ring such that every left module over it has a projective covering (cf. Associative rings and algebras). Right perfect rings are defined similarly. A left perfect ring need not be right perfect.

The following properties of a ring $R$ are equivalent: 1) $R$ is a left perfect ring; 2) every set of pairwise orthogonal idempotents of $R$ is finite, and every non-zero right $R$-module has a non-zero socle; 3) $R$ satisfies the minimum condition for principal right ideals; 4) $R$ satisfies the minimum condition for finitely-generated right ideals; 5) every right $R$-module satisfies the minimum condition for finitely-generated submodules; 6) the Jacobson radical $J$ of $R$ vanishes on the right (that is, for any sequence $a_1,a_2,\dots,$ of elements of $J$ there is an integer $n$ such that the product $a_1\dots a_n=0$), and the quotient ring $R/J$ is Artinian semi-simple; 7) every flat left $R$-module is projective; 8) $R$ contains idempotents $e_1,\dots,e_n$ such that $\sum_{i=1}^ne_i=1$, $e_ie_j=0$ for $i\neq j$, and $e_iRe_i$ is a local ring for every $i$; 9) every left $R$-module satisfies the maximum condition for cyclic submodules; 10) for every $n$ every left $R$-module satisfies the maximum condition for $n$-generated submodules; and 11) every projective left $R$-module has a decomposition with respect to which every direct summand has a complement (see Krull–Remak–Schmidt theorem).

A ring of matrices over a perfect ring is perfect. Idempotent ideals of a perfect ring are generated by idempotents that are central modulo the radical. A group ring $RG$ (see Group algebra) is perfect if and only if $R$ is a perfect ring and $G$ is a finite group. The endomorphism ring of an Abelian group $A$ is perfect only when $A$ is the direct sum of a finite group and finitely many copies of the additive group of rational numbers. Local perfect rings are characterized by the fact that any linearly independent subset of any free left module over it can be extended to a base. The following properties are also equivalent: a) $R$ is a perfect ring and all its quotient rings are self-injective (cf. Self-injective ring); b) every quotient ring of $R$ is quasi-Frobenius (cf. Quasi-Frobenius ring); c) every quotient ring of $R$ has a cogenerator; and d) $R$ is uniserial (cf. Uniserial ring).

References

[1] F. Kasch, "Modules and rings" , Acad. Press (1982) (Translated from German)
[2] C. Faith, "Algebra" , 1–2 , Springer (1973–1979)
[3] V.T. Markov, A.V. Mikhalev, L.A. Skornyakov, A.A. Tuganbaev, "Modules" J. Soviet Math. , 23 : 6 (10983) pp. 2642–2706 Itogi Nauk. Algebra. Topol. Geom. , 19 (1981) pp. 31–134
How to Cite This Entry:
Perfect ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perfect_ring&oldid=33181
This article was adapted from an original article by L.A. Skornyakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article