IDA: Interactive Document on Algebra

Let R be a ring. Addition defines a group structure on R. So every element a has an inverse with respect to the addition. This inverse is denoted -a and is called the opposite of a.

Theorem

The following properties hold for all a, b R.

a * 0 = 0 * a = 0;

a * (-b) = -(ab) = (-a) * b;

(-a) * (-b) = ab;

(-1) * a = -a.