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Ring**

A (unital) ring \((R,+,\times)\) consists of a set \(R\) with binary operations \(+,\times:R\times R\to R\) ('addition' and 'multiplication') such that \((R,+)\) is an abelian group with identity \(0_R\), \((R,\times)\) is a monoid with identity \(1_R\), and \(\times\) distributes over \(+\), i.e \(\forall a,b,c\in R: (a+b)\times c= a\times c+ b\times c, a\times (b+c)=a\times b + a\times c\). One generally also requires that \(0_R\neq 1_R\). Sometimes a ring is not required to have a multiplicative identity, so that \(R,\times\) is instead a semigroup.