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. 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 1 multiplicative identity element, 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

Ring	An algebraic ring.

Functions

absorption	The derived property of two sided absorption (zero).
distributivity	The axiom of two sided distributivity (multiplication).
left_absorption	The derived property of left absorption (zero).
left_distributivity	The axiom of left distributivity (multiplication).
left_negation	The derived property of left negation (multiplicative).
left_zero_divisors	The derived property of left zero divisors.
negation	The derived property of two sided negation (multiplicative).
num_absorption	The derived property of numeric two sided absorption (zero).
num_distributivity	The axiom of numeric two sided distributivity (multiplication).
num_left_absorption	The derived property of numeric left absorption (zero).
num_left_distributivity	The axiom of numeric left distributivity (multiplication).
num_left_negation	The derived property of numeric left negation (multiplicative).
num_left_zero_divisors	The derived property of numeric left zero divisors.
num_negation	The derived property of numeric two sided negation (multiplicative).
num_right_absorption	The derived property of numeric right absorption (zero).
num_right_distributivity	The axiom of numeric right distributivity (multiplication).
num_right_negation	The derived property of numeric right negation (multiplicative).
num_right_zero_divisors	The derived property of numeric right zero divisors.
num_zero_divisors	The derived property of numeric two sided zero divisors.
right_absorption	The derived property of right absorption (zero).
right_distributivity	The axiom of right distributivity (multiplication).
right_negation	The derived property of right negation (multiplicative).
right_zero_divisors	The derived property of right zero divisors.
zero_divisors	The derived property of two sided zero divisors.