Division rings¶
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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]
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Commutative
¶ alias of
sage.categories.fields.Fields
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class
ElementMethods
¶ Bases:
object
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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.
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class
ParentMethods
¶ Bases:
object
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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
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