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

- $R$ is an abelian group with respect to $+$
- $\cdot$ is associative on $R$
- the distributive laws hold, i.e., for all $a,b,c\in R$, $$ a\cdot(b+c) = a\cdot b+a\cdot c \qquad \text{and}\qquad (b+c)\cdot a = b\cdot a+c\cdot a$$
- 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.

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- Review status: reviewed
- Last edited by John Voight on 2020-10-23 11:16:52

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**History:**(expand/hide all)

- 2020-10-23 11:16:52 by John Voight (Reviewed)
- 2020-10-23 11:16:34 by John Voight (Reviewed)
- 2020-10-23 11:16:26 by John Voight
- 2020-10-23 11:15:43 by John Voight
- 2019-04-26 13:57:01 by Holly Swisher
- 2018-08-06 02:38:09 by John Jones

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