Let S , T be rings, is $\displaystyle S \cap T $ and $\displaystyle S \cup T $ also rings?
I think the second one is yes, but the first one is no, but how do I find an example like that? Thanks.
most ring theorists define a subring $\displaystyle S$ of a unitary ring $\displaystyle R$ to be a unitary ring with $\displaystyle 0_S=0_R \neq 1_R=1_S.$ so, to them, $\displaystyle n\mathbb{Z}, \ n \geq 0,$ is a subring of $\displaystyle \mathbb{Z}$ 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 $\displaystyle \begin{bmatrix}a&x&0\\0&b&0\\0&0&c\end{bmatrix}$, and T to be all those of the form $\displaystyle \begin{bmatrix}a&0&0\\0&b&y\\0&0&c\end{bmatrix}$. Then S∪T is not closed under addition.
(I'm sure there are simpler examples that a bona fide ring theorist could provide.)