Rngs

class sage.categories.rngs.Rngs(base_category)

Bases: CategoryWithAxiom_singleton

The category of rngs.

An rng \((S, +, *)\) is similar to a ring but not necessarily unital. In other words, it is a combination of a commutative additive group \((S, +)\) and a multiplicative semigroup \((S, *)\), where \(*\) distributes over \(+\).

EXAMPLES:

Sage

sage: C = Rngs(); C
Category of rngs
sage: sorted(C.super_categories(), key=str)
[Category of associative additive commutative additive associative additive unital distributive magmas and additive magmas,
 Category of commutative additive groups]

sage: sorted(C.axioms())
['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse',
 'AdditiveUnital', 'Associative', 'Distributive']

sage: C is (CommutativeAdditiveGroups() & Semigroups()).Distributive()
True
sage: C.Unital()
Category of rings

Python

>>> from sage.all import *
>>> C = Rngs(); C
Category of rngs
>>> sorted(C.super_categories(), key=str)
[Category of associative additive commutative additive associative additive unital distributive magmas and additive magmas,
 Category of commutative additive groups]

>>> sorted(C.axioms())
['AdditiveAssociative', 'AdditiveCommutative', 'AdditiveInverse',
 'AdditiveUnital', 'Associative', 'Distributive']

>>> C is (CommutativeAdditiveGroups() & Semigroups()).Distributive()
True
>>> C.Unital()
Category of rings

Unital

alias of Rings