# semilocal total quotient ring whose J(R) is not zero

I am interested in rings in which every non-unit is a zero divisor. Can you give me an example of such a ring that also has FINITELY many maximal ideals (semilocal), and whose Jacobson radical is not zero? Thanks

P.S. There is a question about this Rings in which every non-unit is a zero divisor but none of the answers satisfy my constrains.

Edit: I am sorry I haven't specified - I am not interested in an example that is a finite ring, or the one that is local (and both the examples in comments are in this category). Also Artinian rings won't do.

-
Try $k[x]/\langle x^2 \rangle$ where $k$ is a field. –  Konstantin Ardakov May 17 '12 at 13:48
Or any nonreduced finite ring, such as $\mathbb{Z}/n\mathbb{Z}$ if $n$ is not squarefree (this is mentioned in the question referred to). –  Laurent Moret-Bailly May 17 '12 at 14:26
add comment

## 1 Answer

Let $A = k[x_1,x_2, \ldots]$ be the polynomial ring over a field $k$ in infinitely many variables and let $\mathfrak{m} = \langle x_1,x_2,\ldots \rangle$. Then $(A / \mathfrak{m}^2)^2$ satisfies all of your conditions.

-
If I am not mistaken this ring has infinitely many maximal ideals... –  Niki May 19 '12 at 7:46
If $R = (A / \mathfrak{m}^2)^2$ then $J(R) = \mathfrak{m}R$ has square zero, and $R / J(R) \cong (A / \mathfrak{m}) \times (A / \mathfrak{m}) \cong k \times k$. This ring, and hence $R$ itself, has precisely two maximal ideals. –  Konstantin Ardakov May 19 '12 at 12:21
add comment