# Example of a reduced and connected ring that is not a domain?

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.

• What does "connected" mean to you? In my sense something like two lines crossing $k[x,y]/(xy)$ would work. It doesn't fit your last requirement, though. – Hoot Aug 31 '16 at 15:46
• For the first part, a ring of continuous functions from an interval into the real line. – A.G Aug 31 '16 at 15:48
• @Servaes What's a "finite free $\mathbb Z$-algebra"? – user26857 Aug 31 '16 at 15:53

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 $\mathbb Z\times\mathbb Z$.
• @Servaes: Note that if we write $t = x +1$, then this example can be rewritten as $\mathbb Z[t]/t(t-2),$ just as Hoot's example in comments above of $k[x,y]/(xy)$ can be written as $k[s,t]/t(t-s)$, if we set $s = x - y, t = x$. So we can think of the more geometric example as two copies of Spec $k[s]$ glued together at the point $s= 0$, and the number-theoretic example as two copies of Spec $\mathbb Z$ glued together at the point $(2)$ of Spec $\mathbb Z$. So they match under the usual analogy relating $k[s]$ and $\mathbb Z$. – tracing Sep 11 '16 at 3:34