# Thread: Subring of R generated by X

1. ## Subring of R generated by X

Given any ring R AND a subset X of R, we can consider the set of all subrings of R containing X and form their intersection, T, say.

Show that T is the smallest subring of R containing X.

2. ## Re: Subring of R generated by X

Originally Posted by Bernhard
Given any ring R AND a subset X of R, we can consider the set of all subrings of R containing X and form their intersection, T, say.

Show that T is the smallest subring of R containing X.
This is actually a straightforward manipulation of definitions. You'll need to prove two things: $T$ is a subring, then $T$ is the smallest subring containing $X$. Suppose your set of all subrings of $R$ containing $X$ is given by $\mathcal{S}=\{R_\alpha |\alpha \in \Lambda\}$.

For the first one, you really just need to check that $r,s\in T=\displaystyle{\bigcap_{\alpha\in \Lambda} R_{\alpha}}\implies r-s,rs\in T=\displaystyle{\bigcap_{\alpha\in \Lambda} R_{\alpha}}$. The notation is a bit cumbersome, but this fact is actually quite trivial if you think about it. We also have $X\subseteq T$ by definition of $T$.

The second bit is even easier: Let $R_\beta$ be a subring of $R$ containing $X$. Since this is a subring of $R$ containing $X$, we have $R_\beta \in \mathcal{S}$ by definition of $\mathcal{S}$. Using the basic property $A\cap B \Rightarrow A\cap B \subseteq A$ of intersections, this says that $T\subseteq R_\beta$