Definition. By a ring we mean a nonempty set R with two binary operations + and ·, called addition and multiplication (also called product), respectively, such that
(i) (R,+) is an additive abelian group.
(ii) (R, ·) is a multiplicative semigroup.
(iii) Multiplication is distributive (on both sides) over addition; that is, for all ,
(The two distributive laws are respectively called the left distributive law and the right distributive law.)
We usually write ab instead of a · b. The identity of the ...