Multiplicative semi-group

of an associative ring

The semi-group formed by the elements of the given associative ring relative to multiplication. In a unital ring (ring with multiplicative identity) this is a monoid. A non-associative ring is, relative to multiplication, only a magma; it is called the multiplicative system of the ring.

Some properties of a ring can be expressed in terms of the multiplicative semigroup. For example, a factorial ring is one with a multiplicative Gauss semi-group.