Monoid of ideals in a commutative ring¶
WARNING: This is used by some rings that are not commutative!
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'>
>>> 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: 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 - >>> 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: 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 - >>> 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