Ring
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7/7/2026, 9:19:28 AM · Ancient Tree
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A ring is ...1
A ring is a [[Set|set]] $R$ equipped with two operations, usually called addition $+$ and multiplication $\times$, with the following properties :2
1) $R$ is an [[Abelian group|abelian group]] with $+$.3
2) Multiplication is associative: for all $a,b,c\in R$, $(a\times b)\times c= a\times (b\times c)$.4
3) Distributivity: for all $a,b,c\in R$,5
$$a\times (b+c)=a\times b+a\times c \quad \text{and}\quad (a+b)\times c=a\times c+b\times c$$6
4) Multiplicative identity: there exists an element $1$ such that, for all $a\in R$, $1\times a=a\times 1=a$.7
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##### Remarks9
- Some authors do not include the multiplicative identity in the definition. This structure of ring without an identity is often called a rng.Revision 206
7/3/2026, 12:12:33 PM · Ancient Tree
Concept created
A ring is ...