nzmath.ring.Ideal Class Reference

Inheritance diagram for nzmath.ring.Ideal:

Collaboration diagram for nzmath.ring.Ideal:

Public Member Functions
def	__init__ (self, generators, aring)
def	__add__ (self, other)
def	__mul__ (self, other)
def	__eq__ (self, other)
def	__hash__ (self)
def	__ne__ (self, other)
def	__contains__ (self, element)
def	issubset (self, other)
def	issuperset (self, other)
def	reduce (self, element)

Public Attributes
	ring
	generators

Detailed Description

Ideal class is an abstract class to represent the finitely
generated ideals.  Because the finitely-generatedness is not a
restriction for Noetherian rings and in the most cases only
Noetherian rings are used, it is general enough.

Definition at line 460 of file ring.py.

Constructor & Destructor Documentation

__init__()

def nzmath.ring.Ideal.__init__	(		self,
			generators,
			aring
	)

Ideal(generators, ring) creates an ideal of the ring genarated
by the generators.  generators must be an element of the ring
or a list of elements of the ring.

Reimplemented in nzmath.poly.ring.PolynomialIdeal, and nzmath.poly.multiutil.PolynomialIdeal.

Definition at line 468 of file ring.py.

Member Function Documentation

__add__()

def nzmath.ring.Ideal.__add__	(		self,
			other
	)

I + J <=> I.__add__(J)

where I+J = {i+j | i in I and j in J}

Definition at line 484 of file ring.py.

References nzmath.matrix.Matrix.__class__, nzmath.matrix.RingMatrix.__class__, nzmath.matrix.RingSquareMatrix.__class__, nzmath.matrix.FieldMatrix.__class__, nzmath.matrix.MatrixRing.__class__, nzmath.matrix.Subspace.__class__, nzmath.poly.ring.PolynomialIdeal.generators, nzmath.ring.Ideal.generators, nzmath.poly.multiutil.PolynomialIdeal.generators, nzmath.finitefield.FinitePrimeFieldElement.ring, and nzmath.ring.Ideal.ring.

__contains__()

def nzmath.ring.Ideal.__contains__	(		self,
			element
	)

e in I  <=>  I.__contains__(e)

for e in the ring, to which the ideal I belongs.

Reimplemented in nzmath.poly.multiutil.PolynomialIdeal, and nzmath.poly.ring.PolynomialIdeal.

Definition at line 528 of file ring.py.

__eq__()

def nzmath.ring.Ideal.__eq__	(		self,
			other
	)

I == J <=> I.__eq__(J)

Definition at line 508 of file ring.py.

References nzmath.poly.ring.PolynomialIdeal.issubset(), nzmath.ring.Ideal.issubset(), nzmath.poly.ring.PolynomialIdeal.issuperset(), nzmath.ring.Ideal.issuperset(), nzmath.finitefield.FinitePrimeFieldElement.ring, and nzmath.ring.Ideal.ring.

Referenced by nzmath.ring.Ideal.__ne__().

__hash__()

def nzmath.ring.Ideal.__hash__

(

self

)

Definition at line 519 of file ring.py.

__mul__()

def nzmath.ring.Ideal.__mul__	(		self,
			other
	)

I * J <=> I.__mul__(J)

where I*J = {sum of i*j | i in I and j in J}

Definition at line 495 of file ring.py.

References nzmath.matrix.Matrix.__class__, nzmath.matrix.RingMatrix.__class__, nzmath.matrix.RingSquareMatrix.__class__, nzmath.matrix.FieldMatrix.__class__, nzmath.matrix.MatrixRing.__class__, nzmath.matrix.Subspace.__class__, nzmath.poly.ring.PolynomialIdeal.generators, nzmath.ring.Ideal.generators, nzmath.poly.multiutil.PolynomialIdeal.generators, nzmath.finitefield.FinitePrimeFieldElement.ring, and nzmath.ring.Ideal.ring.

__ne__()

def nzmath.ring.Ideal.__ne__	(		self,
			other
	)

I != J <=> I.__ne__(J)

Definition at line 522 of file ring.py.

References nzmath.poly.univar.PolynomialInterface.__eq__(), nzmath.group.Group.__eq__(), nzmath.poly.ratfunc.RationalFunction.__eq__(), nzmath.quad.ReducedQuadraticForm.__eq__(), nzmath.poly.multivar.TermIndices.__eq__(), nzmath.poly.ring.PolynomialRing.__eq__(), nzmath.poly.formalsum.FormalSumContainerInterface.__eq__(), nzmath.ring.Ring.__eq__(), nzmath.factor.misc.FactoredInteger.__eq__(), nzmath.real.Real.__eq__(), nzmath.imaginary.Complex.__eq__(), nzmath.poly.array.ArrayPoly.__eq__(), nzmath.group.GroupElement.__eq__(), nzmath.matrix.Matrix.__eq__(), nzmath.real.RealField.__eq__(), nzmath.intresidue.IntegerResidueClass.__eq__(), nzmath.finitefield.FinitePrimeField.__eq__(), nzmath.imaginary.ComplexField.__eq__(), nzmath.rational.Rational.__eq__(), nzmath.poly.array.ArrayPolyMod.__eq__(), nzmath.ring.RingElement.__eq__(), nzmath.module.Module.__eq__(), nzmath.permute.Permute.__eq__(), nzmath.intresidue.IntegerResidueClassRing.__eq__(), nzmath.algfield.NumberField.__eq__(), nzmath.poly.formalsum.DictFormalSum.__eq__(), nzmath.poly.univar.BasicPolynomial.__eq__(), nzmath.poly.formalsum.ListFormalSum.__eq__(), nzmath.ring.QuotientFieldElement.__eq__(), nzmath.poly.multiutil.PolynomialRingAnonymousVariables.__eq__(), nzmath.poly.ring.RationalFunctionField.__eq__(), nzmath.finitefield.ExtendedFieldElement.__eq__(), nzmath.ring.Ideal.__eq__(), nzmath.rational.RationalField.__eq__(), nzmath.permute.ExPermute.__eq__(), nzmath.permute.PermGroup.__eq__(), nzmath.prime.Zeta.__eq__(), nzmath.prime.FactoredInteger.__eq__(), nzmath.rational.Integer.__eq__(), nzmath.finitefield.ExtendedField.__eq__(), nzmath.poly.univar.SortedPolynomial.__eq__(), nzmath.rational.IntegerRing.__eq__(), and nzmath.matrix.MatrixRing.__eq__().

issubset()

def nzmath.ring.Ideal.issubset	(		self,
			other
	)

Report whether another ideal contains this ideal.

Reimplemented in nzmath.poly.ring.PolynomialIdeal.

Definition at line 536 of file ring.py.

Referenced by nzmath.ring.Ideal.__eq__().

issuperset()

def nzmath.ring.Ideal.issuperset	(		self,
			other
	)

Report whether this ideal contains another ideal.

Reimplemented in nzmath.poly.ring.PolynomialIdeal.

Definition at line 544 of file ring.py.

Referenced by nzmath.ring.Ideal.__eq__().

reduce()

def nzmath.ring.Ideal.reduce	(		self,
			element
	)

Reduce an element with the ideal to simpler representative.

Reimplemented in nzmath.poly.ring.PolynomialIdeal.

Definition at line 552 of file ring.py.

Member Data Documentation

generators

nzmath.ring.Ideal.generators

Definition at line 478 of file ring.py.

Referenced by nzmath.ring.Ideal.__add__(), and nzmath.ring.Ideal.__mul__().

ring

nzmath.ring.Ideal.ring

Definition at line 476 of file ring.py.

Referenced by nzmath.ring.Ideal.__add__(), nzmath.poly.ring.PolynomialIdeal.__contains__(), nzmath.poly.multiutil.PolynomialIdeal.__contains__(), nzmath.ring.Ideal.__eq__(), nzmath.ring.Ideal.__mul__(), nzmath.poly.ring.PolynomialIdeal.__nonzero__(), nzmath.poly.multiutil.PolynomialIdeal.__nonzero__(), nzmath.poly.ring.PolynomialIdeal.__repr__(), nzmath.poly.multiutil.PolynomialIdeal.__repr__(), nzmath.poly.ring.PolynomialIdeal.__str__(), nzmath.poly.multiutil.PolynomialIdeal.__str__(), nzmath.poly.ring.PolynomialIdeal._euclidean_reduce(), nzmath.ring.ResidueClassRing._getOne(), nzmath.ring.ResidueClassRing._getZero(), nzmath.poly.ring.PolynomialIdeal._normalize_generators(), and nzmath.poly.ring.PolynomialIdeal.reduce().

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The documentation for this class was generated from the following file:

• nzmath/ring.py