Noetherian rings¶
EXAMPLES:
sage: from sage.categories.noetherian_rings import NoetherianRings
sage: GF(4, "a") in NoetherianRings() # needs sage.rings.finite_rings
True
sage: QQ in NoetherianRings()
True
sage: ZZ in NoetherianRings()
True
sage: IntegerModRing(4) in NoetherianRings()
True
sage: IntegerModRing(5) in NoetherianRings()
True
>>> from sage.all import *
>>> from sage.categories.noetherian_rings import NoetherianRings
>>> GF(Integer(4), "a") in NoetherianRings() # needs sage.rings.finite_rings
True
>>> QQ in NoetherianRings()
True
>>> ZZ in NoetherianRings()
True
>>> IntegerModRing(Integer(4)) in NoetherianRings()
True
>>> IntegerModRing(Integer(5)) in NoetherianRings()
True
- class sage.categories.noetherian_rings.NoetherianRings[source]¶
Bases:
CategoryThe category of Noetherian rings.
A Noetherian ring is a commutative ring in which every ideal is finitely generated.
See Wikipedia article Noetherian ring
EXAMPLES:
sage: from sage.categories.noetherian_rings import NoetherianRings sage: C = NoetherianRings(); C Category of noetherian rings sage: sorted(C.super_categories(), key=str) [Category of commutative rings]
>>> from sage.all import * >>> from sage.categories.noetherian_rings import NoetherianRings >>> C = NoetherianRings(); C Category of noetherian rings >>> sorted(C.super_categories(), key=str) [Category of commutative rings]
- class ParentMethods[source]¶
Bases:
object- is_noetherian(proof=True)[source]¶
Return
True, since this in an object of the category of Noetherian rings.EXAMPLES:
sage: ZZ.is_noetherian() True sage: QQ.is_noetherian() True sage: ZZ['x'].is_noetherian() True sage: R.<x> = PolynomialRing(QQ) sage: R.is_noetherian() True sage: L.<z> = LazyLaurentSeriesRing(QQ) # needs sage.combinat sage: L.is_noetherian() # needs sage.combinat True
>>> from sage.all import * >>> ZZ.is_noetherian() True >>> QQ.is_noetherian() True >>> ZZ['x'].is_noetherian() True >>> R = PolynomialRing(QQ, names=('x',)); (x,) = R._first_ngens(1) >>> R.is_noetherian() True >>> L = LazyLaurentSeriesRing(QQ, names=('z',)); (z,) = L._first_ngens(1)# needs sage.combinat >>> L.is_noetherian() # needs sage.combinat True
- super_categories()[source]¶
EXAMPLES:
sage: from sage.categories.noetherian_rings import NoetherianRings sage: NoetherianRings().super_categories() [Category of commutative rings]
>>> from sage.all import * >>> from sage.categories.noetherian_rings import NoetherianRings >>> NoetherianRings().super_categories() [Category of commutative rings]