Associative rings and algebras Definition

Perfect ring

Builds on Ring Jacobson radical Semilocal ring

Definition

Let $R$ be a ring with identity and $J = J(R)$ its Jacobson radical. A subset $S \subseteq R$ is left T-nilpotent if for every sequence $a_1, a_2, a_3, \ldots$ of elements of $S$ there is $n \ge 1$ with $a_1 a_2 \cdots a_n = 0$ . $R$ is left perfect if the following hold.

$R$ is semilocal, that is, $R/J$ is a semisimple ring.
$J$ is left T-nilpotent.

Right perfect rings are defined with right T-nilpotent, which asks for $a_n \cdots a_2 a_1 = 0$ instead.

Remarks

Example: a left Artinian ring is left and right perfect: $R/J$ is semisimple, and $J^n = 0$ for some $n$ , so every product of $n$ elements of $J$ is $0$ . Non-example: $\Z_{(p)}$ is semilocal, but $J = p\Z_{(p)}$ is not left T-nilpotent, since the constant sequence $p, p, p, \ldots$ has all products $p^n \ne 0$ . By Bass's theorem P, $R$ is left perfect if and only if every left $R$ -module has a projective cover. This is also equivalent to the descending chain condition on principal right ideals. Many authors take one of these equivalent conditions as the definition.

Reading path

To understand this page from the ground up, read these 6 pages in order. The foundations come first.

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Group concept
Jacobson radical concept
Semisimple ring concept
Ring definition
Semilocal ring definition
Perfect ring definition

Where this leads

1 page builds on this one.

Bass's theorem P

Tag	`BJMEXPKRYM`
URI	`m2://BJMEXPKRYM`
Kind	definition
Writing	W0 stub
MSC	16L30
Defines	Perfect ring
Uses	Ring, Jacobson radical, Semilocal ring

Perfect ring

Definition

Remarks

Reading path

Where this leads

Further reading

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