Quotient Ring Elements

AUTHORS:

  • William Stein
class sage.rings.quotient_ring_element.QuotientRingElement(parent, rep, reduce=True)

Bases: sage.structure.element.RingElement

An element of a quotient ring \(R/I\).

INPUT:

  • parent - the ring \(R/I\)
  • rep - a representative of the element in \(R\); this is used as the internal representation of the element
  • reduce - bool (optional, default: True) - if True, then the internal representation of the element is rep reduced modulo the ideal \(I\)

EXAMPLES:

sage: R.<x> = PolynomialRing(ZZ)
sage: S.<xbar> = R.quo((4 + 3*x + x^2, 1 + x^2)); S
Quotient of Univariate Polynomial Ring in x over Integer Ring by the ideal (x^2 + 3*x + 4, x^2 + 1)
sage: v = S.gens(); v
(xbar,)
sage: loads(v[0].dumps()) == v[0]
True
sage: R.<x,y> = PolynomialRing(QQ, 2)
sage: S = R.quo(x^2 + y^2); S
Quotient of Multivariate Polynomial Ring in x, y over Rational Field by the ideal (x^2 + y^2)
sage: S.gens()
(xbar, ybar)

We name each of the generators.

sage: S.<a,b> = R.quotient(x^2 + y^2)
sage: a
a
sage: b
b
sage: a^2 + b^2 == 0
True
sage: b.lift()
y
sage: (a^3 + b^2).lift()
-x*y^2 + y^2
is_unit()

Return True if self is a unit in the quotient ring.

TODO: This is not fully implemented, as illustrated in the example below. So far, self is determined to be unit only if its representation in the cover ring \(R\) is also a unit.

EXAMPLES:

sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(1 - x*y); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: a*b
1
sage: a.is_unit()
Traceback (most recent call last):
...
NotImplementedError
sage: S(1).is_unit()
True
lc()

Return the leading coefficent of this quotient ring element.

EXAMPLE:

sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex')
sage: I = sage.rings.ideal.FieldIdeal(R)
sage: Q = R.quo( I )
sage: f = Q( z*y + 2*x )
sage: f.lc()
2

TESTS:

sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: (a+3*a*b+b).lc()
3
lift()

If self is an element of \(R/I\), then return self as an element of \(R\).

EXAMPLES:

sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: a.lift()
x
sage: (3/5*(a + a^2 + b^2)).lift()
3/5*x
lm()

Return the leading monomial of this quotient ring element.

EXAMPLE:

sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex')
sage: I = sage.rings.ideal.FieldIdeal(R)
sage: Q = R.quo( I )
sage: f = Q( z*y + 2*x )
sage: f.lm()
xbar

TESTS:

sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: (a+3*a*b+b).lm()
a*b
lt()

Return the leading term of this quotient ring element.

EXAMPLE:

sage: R.<x,y,z>=PolynomialRing(GF(7),3,order='lex')
sage: I = sage.rings.ideal.FieldIdeal(R)
sage: Q = R.quo( I )
sage: f = Q( z*y + 2*x )
sage: f.lt()
2*xbar

TESTS:

sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: (a+3*a*b+b).lt()
3*a*b
monomials()

Return the monomials in self.

OUTPUT:

A list of monomials.

EXAMPLES:

sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: a.monomials()
[a]
sage: (a+a*b).monomials()
[a*b, a]
sage: R.zero().monomials()
[]
reduce(G)

Reduce this quotient ring element by a set of quotient ring elements G.

INPUT:

  • G - a list of quotient ring elements

EXAMPLE:

sage: P.<a,b,c,d,e> = PolynomialRing(GF(2), 5, order='lex')
sage: I1 = ideal([a*b + c*d + 1, a*c*e + d*e, a*b*e + c*e, b*c + c*d*e + 1])
sage: Q = P.quotient( sage.rings.ideal.FieldIdeal(P) )
sage: I2 = ideal([Q(f) for f in I1.gens()])
sage: f = Q((a*b + c*d + 1)^2  + e)
sage: f.reduce(I2.gens())
ebar
variables()

Return all variables occurring in self.

OUTPUT:

A tuple of linear monomials, one for each variable occurring in self.

EXAMPLES:

sage: R.<x,y> = QQ[]; S.<a,b> = R.quo(x^2 + y^2); type(a)
<class 'sage.rings.quotient_ring_element.QuotientRing_generic_with_category.element_class'>
sage: a.variables()
(a,)
sage: b.variables()
(b,)
sage: s = a^2 + b^2 + 1; s
1
sage: s.variables()
()
sage: (a+b).variables()
(a, b)

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