Module un_algebra::ring::ring

Algebraic ring traits.

An algebraic ring R, is an additive commutative group and a multiplicative monoid, and therefore has both addition + and multiplication × operators.

Because of their additive group aspect, rings have a unique 0 additive identity element. Not all authors require rings to have a 1 multiplicative identity element, but in un_algebra they do. This inclusion means un_algebra rings are also termed rings with unity.

In addition to group and monoid axioms ring multiplication is required to distribute over addition.

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

NumRing	A "numeric" algebraic ring.
Ring	An algebraic ring.