Module scirust::algebra::structure::ring
[−]
[src]
Defines the ring algebraic structure.
Briefly, a ring is an abelian group with a second binary operation
that is distributive over the abelian group operation and is associative.
The abelian group operation is called addition
`addition
`additionand the second binary operation is called
multiplication`` in analogy with the integers.
One familiar example of a ring is the set of integers.
A ring is a set R equipped with binary operations + and * satisfying the following eight axioms, called the ring 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 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]
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:
Traits
Ring |
Ring with full equivalence |
RingPartial |
Ring with partial equivalence |