Let $A$ be a commutative ring with $1$. Suppose $R$ and $S$ are two subrings of $A$ containing the same multiplicative unity. Then what is the description of the smallest ring containing $R$ and $S$ inside $A$ ?
In addition if $R$ and $S$ are domains can we say that the smallest ring containing them is also a domain?
I need some explanations to this. Thank you.
The smallest ring containing both $R$ and $S$ is the set $$ \left\{\sum_{i=1}^n r_is_i \ \Bigg| \ n\in \mathbb{N}, r_i\in R, s_i \in S\right\}. $$
Even if $R$ and $S$ are domains, this ring does not have to be a domain: take $A=\mathbb{Z}[x,y]/(xy)$, $R= \mathbb{Z}[x]$ and $S = \mathbb{Z}[y]$.