TITLE: Example of a reduced and connected ring that is not a domain?
QUESTION [5 upvotes]: I'm curious to see an example of a commutative ring with unit that is reduced and connected, but is not a domain. It seems this should be a ring with a trivial nilradical, but nontrivial Jacobson radical. I have no doubt such a ring exists, but I have no clue where to look for one. I'm particularly interested in rings whose additive group is a free and finitely generated abelian group.
Edit: As the comments show, examples of reduced and connected rings that are not domains are plentiful. I'm still curious to see an example that is a finite free $\Bbb{Z}$-algebra, if such a ring exists.

REPLY [6 votes]: How about $\mathbb Z[x]/(x^2-1)$?
Actually, it is the only DaRT entry I have for that type of ring, and it happens to have underlying group isomorphic to $\mathbb Z\times\mathbb Z$.