most ring theorists define a subring of a unitary ring to be a unitary ring with so, to them, is a subring of iff n = 1.
Edit: ok, for some weird reasons, i thought S and T have to be subrings of a unitary ring! lol the problem is just asking for any rings, which makes things much
easier and Opalg's example is perfectly fine! my apologies Opalg!
Well I'm not a ring theorist and nobody told me they had to be unitary.
If you want unitary subrings, how about looking at subrings of the 3×3 matrices over the integers? You could take S to be all those of the form , and T to be all those of the form . Then S∪T is not closed under addition.
(I'm sure there are simpler examples that a bona fide ring theorist could provide.)