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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.

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    $\begingroup$ 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. $\endgroup$ – Hoot Aug 31 '16 at 15:46
  • $\begingroup$ For the first part, a ring of continuous functions from an interval into the real line. $\endgroup$ – A.G Aug 31 '16 at 15:48
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    $\begingroup$ @Servaes What's a "finite free $\mathbb Z$-algebra"? $\endgroup$ – user26857 Aug 31 '16 at 15:53
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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 $\mathbb Z\times\mathbb Z$.

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    $\begingroup$ No problem. Actually, doing your search alerted me to a bug in one of the views. So far I haven't seen anything wrong with the "hits" but the "weak hits" in that search were wrong at the time of this post. That doesn't affect the answer here, of course. $\endgroup$ – rschwieb Sep 1 '16 at 10:23
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    $\begingroup$ @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 $\mathbb Z$. $\endgroup$ – tracing Sep 11 '16 at 3:34

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