Definition:Generated Ring Extension/Smallest Subring

Definition

Let $S$ be a commutative rings with unity.

Let $R$ be a subring of $S$ with unity such that the unity of $R$ is the unity of $S$.

That is, $S$ is a ring extension of $R$.

Let $T \subseteq S$ be a subset of $S$.

The ring extension $R \sqbrk T$ generated by $T$ is the smallest subring of $S$ containing $T$ and $R$, that is, the intersection of all subrings of $S$ containing $T$ and $R$.

Thus $T$ is a generator of $R \sqbrk T$ if and only if $R \sqbrk T$ has no proper subring containing $T$ and $R$.

Also see

• Equivalence of Definitions of Generated Ring Extension