A ring is a set R, together with two binary operations $+, \cdot : R\times R \to R$ that satisfy
- $(R,+)$ is an abelian group
- Associativity
- Distributivity
- Multiplicative identity so $\exists 1_R \in R$ such that $1a = a1 = a\, \, \forall \, a \in R$
Does this mean that the group is only closed under the '$+$' opearation and not the '$\cdot$' one?