[[Ring extension]]
# Adjunction of a ring

Let $R \leq T$ be a [[subring]] and $S \sube T$ be an arbitrary set of elements.
The ring $R[S]$ of $R$ **adjoin** $S$ is the ring generated by $R \cup S$,
i.e. the smallest possible subring of $T$ containing $R \cup S$. #m/def/ring

- If $R[S]$ is an [[integral domain]], then its [[field of fractions]] is denoted $R(S)$.
- The same notation can be used for adjoining a single element, e.g. $R[a]$ or $R(a)$.
- If the ambient ring $T$ is left unspecified and $S$ is a [[monoid]], one considers the [[Monoid ring]].
- See also [[polynomial ring]]
- [[Category ring]]
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