A definition of ring
Topics
About: ring
The table of contents of this article
Starting Context
- The reader knows a definition of set.
- The reader knows a definition of abelian group.
- The reader knows a definition of monoid.
Target Context
- The reader will have a definition of ring.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Definition
Any set, S, with the operations, addition, +, and multiplication, \(\bullet\), that satisfies these conditions: 1) S is an abelian group under addition 2) S is a monoid under multiplication 3) multiplication is distributive with respect to addition, which means that \(p_1 \bullet (p_2 + p_3) = (p_1 \bullet p_2) + (p_1 \bullet p_3)\) and \((p_1 + p_2) \bullet p_3 = (p_1 \bullet p_3) + (p_2 \bullet p_3)\)
2: Note
The existence of the multiplicative identity is supposed here included in being a monoid, although some people seem to define 'ring' without the requirement for the multiplicative identity; such people will call ring by this article "ring with identity".
