Group and Ring

Group

A group is defined to be a monoid in which every element is invertible. Hence a group G is a set G together with a binary operation GxG->G, written (a,b)->ab, such that this

(i)       operation is associative;

(ii)     there is an element uEG with ua=a=au for all aEG;

(iii)   for this element u, there is to each element aEG an element a'EG with aa'=u=a'a.

Ring

A ring R=(R,+,.,1) is a set R with two binary operations, addition and multiplication, and a nullary operation, “select 1”, such that

1. (R,+) is an abelian group under addition;
2. (R,.,1) is a monoid under multiplication;
3. Multiplication is distributive (on both sides) over addition.

The last requirement means that all triples of elements a,b,c in R satisfy the identities

a(b+c)=ab+ac,  (a+b)c=ac+bc