Monoid of ideals in a commutative ring#

WARNING: This is used by some rings that are not commutative!

Sage

sage: MS = MatrixSpace(QQ, 3, 3)                                                    # needs sage.modules
sage: type(MS.ideal(MS.one()).parent())                                             # needs sage.modules
<class 'sage.rings.ideal_monoid.IdealMonoid_c_with_category'>

Python

>>> from sage.all import *
>>> MS = MatrixSpace(QQ, Integer(3), Integer(3))                                                    # needs sage.modules
>>> type(MS.ideal(MS.one()).parent())                                             # needs sage.modules
<class 'sage.rings.ideal_monoid.IdealMonoid_c_with_category'>

sage.rings.ideal_monoid.IdealMonoid(R)[source]#

Return the monoid of ideals in the ring R.

EXAMPLES:

Sage

sage: R = QQ['x']
sage: from sage.rings.ideal_monoid import IdealMonoid
sage: IdealMonoid(R)
Monoid of ideals of Univariate Polynomial Ring in x over Rational Field

Python

>>> from sage.all import *
>>> R = QQ['x']
>>> from sage.rings.ideal_monoid import IdealMonoid
>>> IdealMonoid(R)
Monoid of ideals of Univariate Polynomial Ring in x over Rational Field

class sage.rings.ideal_monoid.IdealMonoid_c(R)[source]#

Bases: Parent

The monoid of ideals in a commutative ring.

Element[source]#: alias of Ideal_generic

ring()[source]#

Return the ring of which this is the ideal monoid.

EXAMPLES:

Sage

sage: R = QuadraticField(-23, 'a')                                          # needs sage.rings.number_field
sage: from sage.rings.ideal_monoid import IdealMonoid
sage: M = IdealMonoid(R); M.ring() is R                                     # needs sage.rings.number_field
True

Python

>>> from sage.all import *
>>> R = QuadraticField(-Integer(23), 'a')                                          # needs sage.rings.number_field
>>> from sage.rings.ideal_monoid import IdealMonoid
>>> M = IdealMonoid(R); M.ring() is R                                     # needs sage.rings.number_field
True