Monoid of ideals in a commutative ring - General Rings, Ideals, and Morphisms

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# 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)

  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)

  Bases: `Parent`

  The monoid of ideals in a commutative ring.

  Element

    alias of `Ideal_generic`

  ring()

    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