Module scirust::algebra::structure::commutative_ring

Defines the commutative ring algebraic structure.

A commutative ring is a ring where the multiplication operation is commutative.

A commutative ring is a set R equipped with binary operations + and * satisfying the following nine axioms:

R is an abelian group under addition, meaning:

• (a + b) + c = a + (b + c) for all a, b, c in R (+ is associative).
• There is an element 0 in R such that a + 0 = a and 0 + a = a for all a in R (0 is the additive identity).
• For each a in R there exists −a in R such that a + (−a) = (−a) + a = 0 (−a is the additive inverse of a).
• a + b = b + a for all a, b in R (+ is commutative).

R is a commutative monoid under multiplication, meaning:

• (a * b) * c = a * (b * c) for all a, b, c in R (* is associative).
• There is an element 1 in R such that a * 1 = a and 1 * a = a for all a in R (1 is the multiplicative identity).[2]
• a * b = b * a for all a, b in R (* is commutative).

Multiplication distributes over addition:

• a * (b + c) = (a * b) + (a * c) for all a, b, c in R (left distributivity).
• (b + c) * a = (b * a) + (c * a) for all a, b, c in R (right distributivity).

References:

• http://en.wikipedia.org/wiki/Algebraic_structure
• http://en.wikipedia.org/wiki/Group_(mathematics)
• http://en.wikipedia.org/wiki/Monoid
• http://en.wikipedia.org/wiki/Ring_(mathematics)

Traits

CommutativeRing	Commutative ring with full equivalence
CommutativeRingPartial	Commutative ring with partial equivalence