TITLE: How to construct the subring generated by a set, T?
QUESTION [7 upvotes]: I'm trying to find a constructive way of describing the subring generated by some subset, T, of a ring R.  I think I could describe it as all finite sums of finite products of elements of T, but I have no idea how to write that as a set or how to prove that it is equal to the subring generated by T.

REPLY [2 votes]: Let $S$ be a subset of a commutative ring $R$. By the universal property of polynomial rings, there exists a (unique) morphism of $\mathbb{Z}-$algebras $\phi:\mathbb{Z}[x_{s}: s\in S]\rightarrow R$ such that $\phi(x_{s})=s$ for all $s\in S$. Then $\mathbb{Z}[S]$, the image of this morphism, is the subring of $R$ which is generated by $S$. In fact, it is easy to see that $\mathbb{Z}[S]$ is the intersection of all subrings of $R$ that are contain $S$. One can observe that $a\in\mathbb{Z}[S]$ if and only if $a=f(s_{1},...,s_{n})$ where $f(x_{s_{1}},...,x_{s_{n}})\in\mathbb{Z}[x_{s}: s\in S]$ and $s_{i}\in S$ for all $i$.