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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.

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    $\begingroup$ Try $k[x]/\langle x^2 \rangle$ where $k$ is a field. $\endgroup$
    – user91132
    Commented May 17, 2012 at 13:48
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    $\begingroup$ 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). $\endgroup$
    – Laurent Moret-Bailly
    Commented May 17, 2012 at 14:26

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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.

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  • $\begingroup$ If I am not mistaken this ring has infinitely many maximal ideals... $\endgroup$
    – Niki
    Commented May 19, 2012 at 7:46
  • $\begingroup$ 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. $\endgroup$
    – user91132
    Commented May 19, 2012 at 12:21

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