Terminology of "algebraically closed rings"

Question

There are various approaches how to generalize the notion of an algebraically closed field to the context of commutative rings. A good survey is R. Raphael, On algebraic closures. I am interested in the following notion of an "algebraically closed" commutative ring $R$, which is not exactly covered there:

Every monic polynomial over $R$ decomposes as a product product of linear factors.

This is related to the notion studied in E. Enochs, Totally integrally closed rings, but not identical to it (in the non-reduced case).

Question. Is there an established name for commutative rings satisfying the property above? Can you name some literature where these rings have been studied?

Notice that I am not interested in literature about related notions (Raphael's paper mentions a couple of them), I am asking specifically about this notion.

Let me mention the following existence result: Every commutative ring $R$ admits a tight integral extension $R \hookrightarrow S$ such that $S$ is "algebraically closed" in the sense above. (Notice that this result holds in Enochs's setting only when $R$ is reduced.)

Just wanted to add that these are called "absolutely integrally closed" rings in the Stacks Project Tag 0DCK. — Minseon Shin, Feb 10 '22 at 09:30
Related prior question (also here and here). It would be helpful to add a definition of a "tight" extension. — Bill Dubuque, Feb 10 '22 at 09:47
Thank you! The term "absolutely integrally closed" is also used in the book "Integral Closure of Ideals, Rings, and Modules" by Swanson-Huneke in Bill's link. You may post this as an answer, I will accept it. — Martin Brandenburg, Feb 10 '22 at 11:36

score 1 · Accepted Answer · answered Nov 06 '22 at 22:53

1

The question has been answered in the comments. They are called "absolutely integrally closed" rings.

PS: I have posted a follow-up question here.

answered Nov 06 '22 at 22:53

Martin Brandenburg

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Terminology of "algebraically closed rings"

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