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?
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Duchamp Gérard H. E.
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E.R
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I’ve never seen anything I would call analogous for semilocal rings. – rschwieb Sep 01 '19 at 21:24
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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.
rschwieb
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For proving things about a semi-local ring $R$, the following local-global property can be useful:
If a polynomial $f \in R[x_1, \ldots, x_n]$ represents a unit over $R_P$ for every maximal ideal $P$ of $R$, then $f$ represents a unit over $R$.
This property holds for semi-local rings and a few others, such as the ring of all algebraic integers. A good reference for consequences of this property as well as some similar properties is the paper
- Estes, Dennis R., and Robert M. Guralnick. "Module equivalences: local to global when primitive polynomials represent units." Journal of Algebra 77.1 (1982): 138-157. https://www.sciencedirect.com/science/article/pii/0021869382902824
As an example of how this hypothesis may be used, this class of rings has a primitive element theorem and normal basis theorem, see
- Paques, A. "On the primitive element and normal basis theorems." Communications in Algebra 16.3 (1988): 443-455. https://www.tandfonline.com/doi/abs/10.1080/00927878808823581?journalCode=lagb20
Skip
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See also Neher, Gille, "Group schemes over LG-rings and applications to cancellation theorems and Azumaya algebras", https://arxiv.org/abs/2407.02021 – Skip Sep 04 '24 at 02:21