An [abelian group,](Abelian%20groups.md) $\langle R,+,\rangle,$ equipped with an operator $+,$ is also a _ring,_ $\langle R,+,\cdot\rangle$ if it is also equipped with an additional operator $\cdot,$ such that for $\cdot,$ 1. $(r\cdot s)\cdot t = r \cdot (s\cdot t) \,\,\,\, \forall r,s,t \in R$ (associativity) 2. there exists an element $1_R$ such that $r\cdot1_R=1_R\cdot r \,\,\,\, \forall r\in R$ (two sided identity) As a matter of convention, sometimes the multiplicative identity is excluded from the definition of a "ring" and we may instead refer to a ring with the two sided identity as a "ring with $1_R$." # Abelian groups that form rings For an [abelian group,](Abelian%20groups.md) $\langle R,+,\rangle,$ equipped with an operator $+,$ the corresponding identity element is $0_R,$ which may also simply be denoted as $0.$ # [zero rings](Zero%20ring.md)  # [[Commutative rings]] #MathematicalFoundations/Algebra/AbstractAlgebra/RingTheory #MathematicalFoundations/Numbers