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: Category

The 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 ElementMethods[source]

Bases: object

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]