Rings

sage.categories.rings.Rings

The category of rings

Associative rings with unit, not necessarily commutative

EXAMPLES:

sage: Rings()
Category of rings
sage: sorted(Rings().super_categories(), key=str)
[Category of rngs, Category of semirings]

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

sage: Rings() is (CommutativeAdditiveGroups() & Monoids()).Distributive()
True
sage: Rings() is Rngs().Unital()
True
sage: Rings() is Semirings().AdditiveInverse()
True

Todo

(see: http://trac.sagemath.org/sage_trac/wiki/CategoriesRoadMap)

  • Make Rings() into a subcategory or alias of Algebras(ZZ);
  • A parent P in the category Rings() should automatically be in the category Algebras(P).