Ivan Shishkin, Birch Grove

Ring

Concept history

A revision trail for this concept page.

4 revisions

Back

Revision 830

8/2/2026, 7:43:05 AM · Sequoia

Concept edited

Recorded titleRing
Recorded typeDefinition
Compare with revision 8282 changed lines
1A ring is a [[Set|set]] $R$ equipped with two operations, usually called addition $+$ and multiplication $\times$, with the following properties :
1A ring is a [[Set|set]] $R$ equipped with two [[Operation|binary operations]], usually called addition $+$ and multiplication $\times$, with the following properties :
21) $R$ is an [[Abelian group|abelian group]] with $+$.
32) Multiplication is associative: for all $a,b,c\in R$, $(a\times b)\times c= a\times (b\times c)$.
43) Distributivity: for all $a,b,c\in R$,
5$$a\times (b+c)=a\times b+a\times c \quad \text{and}\quad (a+b)\times c=a\times c+b\times c$$
64) Multiplicative identity: there exists an element $1$ such that, for all $a\in R$, $1\times a=a\times 1=a$.
7
8We then denote $R$ by $(R,+,\times)$.
9
10##### Remarks
11- Some authors do not include the multiplicative identity in the definition. This structure of ring without an identity is often called a rng.

Revision 828

8/2/2026, 7:32:59 AM · Sequoia

Concept edited

Compare with revision 2532 changed lines
1A ring is a [[Set|set]] $R$ equipped with two operations, usually called addition $+$ and multiplication $\times$, with the following properties :
21) $R$ is an [[Abelian group|abelian group]] with $+$.
32) Multiplication is associative: for all $a,b,c\in R$, $(a\times b)\times c= a\times (b\times c)$.
43) Distributivity: for all $a,b,c\in R$,
5$$a\times (b+c)=a\times b+a\times c \quad \text{and}\quad (a+b)\times c=a\times c+b\times c$$
64) Multiplicative identity: there exists an element $1$ such that, for all $a\in R$, $1\times a=a\times 1=a$.
7
8We then denote $R$ by $(R,+,\times)$.
9
8##### Remarks
9- Some authors do not include the multiplicative identity in the definition. This structure of ring without an identity is often called a rng.

Revision 253

7/7/2026, 9:19:28 AM · Ancient Tree

Concept edited

Compare with revision 20610 changed lines
1A ring is ...
1A ring is a [[Set|set]] $R$ equipped with two operations, usually called addition $+$ and multiplication $\times$, with the following properties :
21) $R$ is an [[Abelian group|abelian group]] with $+$.
32) Multiplication is associative: for all $a,b,c\in R$, $(a\times b)\times c= a\times (b\times c)$.
43) Distributivity: for all $a,b,c\in R$,
5$$a\times (b+c)=a\times b+a\times c \quad \text{and}\quad (a+b)\times c=a\times c+b\times c$$
64) Multiplicative identity: there exists an element $1$ such that, for all $a\in R$, $1\times a=a\times 1=a$.
7
8##### Remarks
9- 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 ...