View source for Perfect ring

''left''

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]]).

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]]).

====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>