Take the 2-minute tour ×  This may be a simple question: Find
all subrings of $\mathbb{Z}^2$.  share|improve this question Are
your rings required to have an identity, and do your subrings
have to have the same identity? –  Qiaochu Yuan Jan 2 '11 at
13:56 add comment  2 Answers  The argument sketched in Qiaochu's
answer is actually a special case of a general relationship
between subalgebras and congruences. Generalizing the way that
one defines congruences for the ring of integers modulo
$\rm\:m\:$, one defines a congruence $\equiv$ as an equivalence
relation on an algebra $\rm\:A\:$ that is compatible with the
operations on $\rm\:A\:$. For example, for rings such
compatibility means that if $\rm a'\equiv a,\ b'\equiv b\:$ then
$\rm\: a'+b'\:\equiv\: a+b\ $ and similarly for all other
operations. Now, viewing the equivalence relation $\equiv$ as a
subset of $\rm\ A\times A\:,\: $ this compatibility condition is
a closure condition: if $\rm\: (a',a),\:(b',b)\in\;\equiv\:$
then $\rm\: (a'+b',a+b)\in\; \equiv\: