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 http://mathoverflow.net/questions/42647/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.