# Generic implementation of one- and two-sided ideals of non-commutative rings.¶

Generic implementation of one- and two-sided ideals of non-commutative rings.

AUTHOR:

EXAMPLES:

sage: MS = MatrixSpace(ZZ,2,2)
sage: MS*MS([0,1,-2,3])
Left Ideal
(
[ 0  1]
[-2  3]
)
of Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: MS([0,1,-2,3])*MS
Right Ideal
(
[ 0  1]
[-2  3]
)
of Full MatrixSpace of 2 by 2 dense matrices over Integer Ring
sage: MS*MS([0,1,-2,3])*MS
Twosided Ideal
(
[ 0  1]
[-2  3]
)
of Full MatrixSpace of 2 by 2 dense matrices over Integer Ring


See letterplace_ideal for a more elaborate implementation in the special case of ideals in free algebras.

TESTS:

sage: A = SteenrodAlgebra(2)
sage: IL = A*[A.1+A.2,A.1^2]; IL
Left Ideal (Sq(2) + Sq(4), Sq(1,1)) of mod 2 Steenrod algebra, milnor basis
sage: TestSuite(IL).run(skip=['_test_category'],verbose=True)
running ._test_eq() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_pickling() . . . pass

class sage.rings.noncommutative_ideals.IdealMonoid_nc(R)

Base class for the monoid of ideals over a non-commutative ring.

Note

This class is essentially the same as IdealMonoid_c, but does not complain about non-commutative rings.

EXAMPLES:

sage: MS = MatrixSpace(ZZ,2,2)
sage: MS.ideal_monoid()
Monoid of ideals of Full MatrixSpace of 2 by 2 dense matrices over Integer Ring

class sage.rings.noncommutative_ideals.Ideal_nc(ring, gens, coerce=True, side='twosided')

Generic non-commutative ideal.

All fancy stuff such as the computation of Groebner bases must be implemented in sub-classes. See LetterplaceIdeal for an example.

EXAMPLES:

sage: MS = MatrixSpace(QQ,2,2)
sage: I = MS*[MS.1,MS.2]; I
Left Ideal
(
[0 1]
[0 0],

[0 0]
[1 0]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: [MS.1,MS.2]*MS
Right Ideal
(
[0 1]
[0 0],

[0 0]
[1 0]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field
sage: MS*[MS.1,MS.2]*MS
Twosided Ideal
(
[0 1]
[0 0],

[0 0]
[1 0]
)
of Full MatrixSpace of 2 by 2 dense matrices over Rational Field

side()

Return a string that describes the sidedness of this ideal.

EXAMPLES:

sage: A = SteenrodAlgebra(2)
sage: IL = A*[A.1+A.2,A.1^2]
sage: IR = [A.1+A.2,A.1^2]*A
sage: IT = A*[A.1+A.2,A.1^2]*A
sage: IL.side()
'left'
sage: IR.side()
'right'
sage: IT.side()
'twosided'


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Monoid of ideals in a commutative ring

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Homomorphisms of rings