Module un_algebra::ring::ring

Algebraic rings.

An algebraic ring R, is an additive commutative group and a multiplicative monoid, and therefore has both addition + and multiplication × operators. In addition to group and monoid axioms ring multiplication is required to distribute over addition.

Because of their additive group aspect, rings have a unique 0 additive identity element. Not all authors require rings to have a multiplicative identity element (1), but in un_algebra they do, i.e. un_algebra rings are rings with unity.

Axioms

∀x, y, z ∈ R

Distributivity (left): x × (y + z) = (x × y) + (x × z).
Distributivity (right): (x + y) × z = (x × z) + (y × z).

References

See references for a formal definition of a ring.

Traits

NumRingLaws	Laws (axioms and properties) of rings.
Ring	An algebraic ring.
RingLaws	Laws (axioms and properties) of rings.