Difference between revisions of "Perfect ring"
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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 $R$ 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]]). | 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]]). | ||
Latest revision as of 19:59, 30 October 2016
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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 |
Perfect ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Perfect_ring&oldid=33181