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sagemath
GitHub Repository: sagemath/sagelib
 Path: blob/master/sage/categories/commutative_rings.py
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r"""

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Commutative rings

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"""

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#*****************************************************************************

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#  Copyright (C) 2005      David Kohel <[email protected]>

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#                          William Stein <[email protected]>

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#                2008      Teresa Gomez-Diaz (CNRS) <[email protected]>

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#                2008-2009 Nicolas M. Thiery <nthiery at users.sf.net>

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#

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#  Distributed under the terms of the GNU General Public License (GPL)

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#                  http://www.gnu.org/licenses/

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#******************************************************************************

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from sage.categories.category import Category

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from sage.categories.category_singleton import Category_singleton

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from sage.misc.cachefunc import cached_method

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class CommutativeRings(Category_singleton):

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    """

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    The category of commutative rings

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    commutative rings with unity, i.e. rings with commutative * and 

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    a multiplicative identity

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    EXAMPLES::

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      sage: CommutativeRings()

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      Category of commutative rings

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      sage: CommutativeRings().super_categories()

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      [Category of rings]

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    TESTS::

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        sage: TestSuite(CommutativeRings()).run()

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        sage: QQ['x,y,z'] in CommutativeRings()

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        True

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        sage: GroupAlgebra(DihedralGroup(3), QQ) in CommutativeRings()

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        False

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        sage: MatrixSpace(QQ,2,2) in CommutativeRings()

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        False

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    GroupAlgebra should be fixed::

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        sage: GroupAlgebra(CyclicPermutationGroup(3), QQ) in CommutativeRings() # todo: not implemented

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        True

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    """

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    @cached_method

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    def super_categories(self):

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        """

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        EXAMPLES::

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            sage: CommutativeRings().super_categories()

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            [Category of rings]

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        """

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        from sage.categories.rings import Rings

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        return [Rings()]

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    class ParentMethods:

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        def is_commutative(self):

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            """

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            Return True, since commutative rings are commutative.

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            EXAMPLES::

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                sage: Parent(QQ,category=CommutativeRings()).is_commutative()

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                True

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            """

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            return True

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    class ElementMethods:

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        pass

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