TITLE: Isomorphic groups but not isomorphic rings
QUESTION [12 upvotes]: Provide an example of two rings that have the same characteristic, are isomorphic as groups but are not isomorphic as rings.
I'm confused with how to being. I know that having the same characteristic means that the concatenation is the same number to receive the zero element.

REPLY [4 votes]: One surprising class of examples comes from the opposite ring construction: given a ring $R$, the opposite ring $R^{op}$ has the same underlying set and same additive structure, but "backwards" multiplication: writing "$\otimes$" for the $R^{op}$ multiplication and "$\times$" for the $R$ multiplication, the rule is $$a\otimes b=b\times a.$$ Even though this looks the same, we need not have $R\cong R^{op}$, even though they are of course isomorphic as groups and extremely similar as rings; see here.