Definition of a ring. Let us consider a set R of elements a, b, c, …, such that for arbitrary elements a and b of R there is a uniquely defined sum a + b and product ab (sometimes written as a·b) which are also elements of R. The words addition and multiplication, as in the ordinary usage of elementary algebra, will be respectively associated with the operations of forming a sum, or a product, of elements of R. Such a set is said to be a ring if addition and multiplication have the five properties listed below, it being assumed that a, b, and c are arbitrary elements of R, either distinct or identical:

P1 · a + (b + c) = (a + b) + c (associative law of addition);

P2 · a + b = b + a (commutative law of addition);

P3 · The equation a + x = b has a solution x in R;

P4 · a(bc) = (ab)c (associative law of multiplication);

P5 · a(b + c) = ab + ac, (b + c)a = ba + ca (distributive laws).

The importance of the concept of ring follows primarily from the fact that there are so many important mathematical systems of quite different types which are rings according to the above definition. Naturally, what they all have in common are the properties used in the definition of a ring, together with any properties which are logical consequences of these.