1. ## Subring problem

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.

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?
Here's an example to think about. Let S denote all integer multiples of 3, and T denote all integer multiples of 5. Then 3∈S and 5∈T. But 3+5=8 which is not a multiple of 3 or 5. So S∪T is not closed under addition.

3. Originally Posted by Opalg
Here's an example to think about. Let S denote all integer multiples of 3, and T denote all integer multiples of 5. Then 3∈S and 5∈T. But 3+5=8 which is not a multiple of 3 or 5. So S∪T is not closed under addition.
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!

4. Originally Posted by NonCommAlg
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.
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.)

5. Originally Posted by Opalg
(I'm sure there are simpler examples that a bona fide ring theorist could provide.)
$\displaystyle S=R[X], \ T=R[Y],$ as subrings of the polynomial ring $\displaystyle R[X,Y].$ then $\displaystyle S \cup T$ is not closed under addition (or multiplication).