Division rings#
- class sage.categories.division_rings.DivisionRings(base_category)#
Bases:
sage.categories.category_with_axiom.CategoryWithAxiom_singleton
The category of division rings
A division ring (or skew field) is a not necessarily commutative ring where all non-zero elements have multiplicative inverses
EXAMPLES:
sage: DivisionRings() Category of division rings sage: DivisionRings().super_categories() [Category of domains]
- Commutative#
alias of
sage.categories.fields.Fields
- class ElementMethods#
Bases:
object
- Finite_extra_super_categories()#
Return extraneous super categories for
DivisionRings().Finite()
.EXAMPLES:
Any field is a division ring:
sage: Fields().is_subcategory(DivisionRings()) True
This methods specifies that, by Weddeburn theorem, the reciprocal holds in the finite case: a finite division ring is commutative and thus a field:
sage: DivisionRings().Finite_extra_super_categories() (Category of commutative magmas,) sage: DivisionRings().Finite() Category of finite enumerated fields
Warning
This is not implemented in
DivisionRings.Finite.extra_super_categories
because the categories of finite division rings and of finite fields coincide. See the section Deduction rules in the documentation of axioms.
- class ParentMethods#
Bases:
object
- extra_super_categories()#
Return the
Domains
category.This method specifies that a division ring has no zero divisors, i.e. is a domain.
See also
The Deduction rules section in the documentation of axioms
EXAMPLES:
sage: DivisionRings().extra_super_categories() (Category of domains,) sage: "NoZeroDivisors" in DivisionRings().axioms() True