A commutative ring with 1 is called semi-local if it has finitely many maximal ideals and is called local if it has only one maximal ideal. There are some algebraic charactrizations for local rings. For example a ring $R$ is local ring if and only if all elements of $R$ that are not units form an ideal if and only if either $r$ or $1-r$ is unit for all $r\in R$. Is there any such caracterizations for semi-local rings with more than one maximal ideals?

  • $\begingroup$ I’ve never seen anything I would call analogous for semilocal rings. $\endgroup$ – rschwieb Sep 1 '19 at 21:24

I did some thinking and came to the conclusion that an analogous statement seems most likely for the slightly better class of semiperfect rings.

Recall that a ring is a clean ring if every element is the sum of an idempotent and a unit.

Proposition: a ring is local iff it is clean and has trivial idempotents.

You see, if $x$ is not already a unit, so that $x+0$ is such a decomposition, then $(x-1)+1$ works.

I think the analogous statement would be

Proposition: a ring is semiperfect iff it is clean and has a complete orthogonal set of primitive idempotents.

If the ring is commutative, then such a ring is a finite product of local rings, and every element can be expressed as $u+e$ where $e$ is the sum of finitely many elements of the complete orthogonal set.

I could not see a version specifically for semilocal rings. They can be a bit less nice.


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