Definition of ring (reviewed)

A ring is a set $R$ with two binary operations $+$ and $\cdot$ such that

1. $R$ is an abelian group with respect to $+$
2. $\cdot$ is associative on $R$
3. the distributive laws hold, i.e., for all $a,b,c\in R$, $$ a\cdot(b+c) = a\cdot b+a\cdot c \qquad \text{and}\qquad (b+c)\cdot a = b\cdot a+c\cdot a$$
4. there is an identity element with respect to the operation $\cdot$, typically denoted by $1_R$ or, more simply, by $1$.

The identity element of $R$ as a group with respect to $+$ is typically denoted by $0_R$ or, more simply, by $0$.

The ring $R$ is a commutative ring if $R$ is a ring such that the operation $\cdot$ is commutative on $R$.

We say that $R$ is a rng (also called ring without identity) if conditions 1-3 (but not necessarily 4) are satisfied.