Ring extension

Adjunction of a ring

Let 𝑅 ≤𝑇 be a subring and 𝑆 ⊆𝑇 be an arbitrary set of elements. The ring 𝑅[𝑆] of 𝑅 adjoin 𝑆 is the ring generated by 𝑅 ∪𝑆, i.e. the smallest possible subring of 𝑇 containing 𝑅 ∪𝑆. 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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