Adjunction of a ring

Let $𝑅 \leq 𝑇$ be a subring and $𝑆 \subseteq 𝑇$ 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 rng

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