# When is the quotient ring finite?

My question is in the title:

Given a commutative ring $R$ with identity $1\neq 0$ and an ideal $M$ of $R$. For what condition of $M$, the quotient ring $R/M$ is finite?

My question comes from the theorem: if $M$ is a maximal ideal, then $M$ is a prime ideal. In general, the converse is not true. However, we know that if $M$ is a prime ideal then $R/M$ is an integral domain, and if $R/M$ is also finite, then it is a field, and the converse holds.

Thank you so much for any help.

• There isn't really any condition that is simpler than saying "$R/M$ is finite" in general. Is there some particular class of rings $R$ you have in mind? – Eric Wofsey Dec 3 '15 at 16:40
• I did not think of any particular ring. For example, take the ring $\mathbb{Z}[x]$ and the ideal $(p,x)$ generated by $p$ and $x$, where $p$ is a prime number. Then the quotient ring $\mathbb{Z}[x]/(p,x)$ is isomorphic to $\mathbb{Z}/p\mathbb{Z}$, which is finite. – Tien Kha Pham Dec 3 '15 at 16:46
• You mean besides the very evident one of $R=\cup_{a\in A} M$ where $A$ is a finite set of elements? – Zelos Malum Dec 3 '15 at 17:39
• @ZelosMalum Meant to write $R=\bigcup_{a\in A} a+M$ for a finite set $A$, I think? You could repost and then I'll blow this comment away... – rschwieb Dec 3 '15 at 17:51
• That was exactly what I was intending – Zelos Malum Dec 3 '15 at 17:51

## 1 Answer

Speaking from my own experiences, I'm not aware of a nontrivial characterization of when a quotient ring is finite.

Actually, you could generalize your question by asking "When is $$R/P$$ Artinian?" since an Artinian domain is also a field. There is not, as far as I'm aware, any useful characterization of when $$R/I$$ is Artinian.

Rings for which $$R/J(R)$$ is Artinian are called semilocal rings. There is a slightly useful characterization of commutative semilocal rings: they're exactly the commutative rings with finitely many maximal ideals.

This doesn't generalize your original post, but if $$R/P$$ is von Neumann regular, $$R/P$$ is also a field.

So "when is $$R/I$$ von Neumann regular?" is another question, but again there does not seem to be any useful characterization. Rings for which $$R/J(R)$$ is von Neumann regular are called semiregular rings.

Since you seem to be interested in cases where prime ideals turn out to be maximal, I'll link you to a characterization of commutative rings whose prime ideals are all maximal.