# A characterization for semilocal rings

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?

• I’ve never seen anything I would call analogous for semilocal rings. – 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.