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.

of a general magma

If $M$ is a magma with a binary operation written $\cdot$ then the left and right multiplication operators are the maps from $M$ to itself defined by $L_x :y \mapsto x \cdot y$ and $R_x : y \mapsto y \cdot x$. The multiplicative or multiplication semigroup of $(M,{\cdot})$ is the semigroup generated by the operators $L_x$ and $R_x$ within the semigroup of all maps from $M$ to itself.

Properties of the magma $M$ are reflected in those of its multiplication semigroup. For example, the flexible identity is equivalent to each pair $L_x$, $R_x$ commuting, and associativity to all $L_x$ commuting with all $R_y$; $M$ is a quasi-group if each of the left and right multiplication operators are permutations.

References

[a1]	R.H. Bruck, "A survey of binary systems" Ergebnisse der Mathematik und ihrer Grenzgebiete. Neue Folge. 20 Springer (1958) Zbl 0081.01704