def quotient(self, I, names=None):
"""
Quotient of a ring by a two-sided ideal.
INPUT:
- ``I``: A twosided ideal of this ring.
- ``names``: a list of strings to be used as names
for the variables in the quotient ring.
EXAMPLES::
sage: F.<x,y,z> = FreeAlgebra(QQ, 3)
sage: I = F*[x*y+y*z,x^2+x*y-y*x-y^2]*F
sage: Q = Rings().parent_class.quotient(F,I); Q
Quotient of Free Algebra on 3 generators (x, y, z) over Rational Field by the ideal (x*y + y*z, x^2 + x*y - y*x - y^2)
sage: Q.0
xbar
sage: Q.1
ybar
sage: Q.2
zbar
"""
from sage.rings.quotient_ring import QuotientRing
return QuotientRing(self, I, names=names)
def quotient(self, I, names=None):
"""
Quotient of a ring by a two-sided ideal.
INPUT:
- ``I``: A twosided ideal of this ring.
- ``names``: a list of strings to be used as names
for the variables in the quotient ring.
EXAMPLES:
Usually, a ring inherits a method :meth:`sage.rings.ring.Ring.quotient`.
So, we need a bit of effort to make the following example work with the
category framework::
sage: F.<x,y,z> = FreeAlgebra(QQ)
sage: from sage.rings.noncommutative_ideals import Ideal_nc
sage: class PowerIdeal(Ideal_nc):
... def __init__(self, R, n):
... self._power = n
... self._power = n
... Ideal_nc.__init__(self,R,[R.prod(m) for m in CartesianProduct(*[R.gens()]*n)])
... def reduce(self,x):
... R = self.ring()
... return add([c*R(m) for c,m in x if len(m)<self._power],R(0))
...
sage: I = PowerIdeal(F,3)
sage: Q = Rings().parent_class.quotient(F,I); Q
Quotient of Free Algebra on 3 generators (x, y, z) over Rational Field by the ideal (x^3, x^2*y, x^2*z, x*y*x, x*y^2, x*y*z, x*z*x, x*z*y, x*z^2, y*x^2, y*x*y, y*x*z, y^2*x, y^3, y^2*z, y*z*x, y*z*y, y*z^2, z*x^2, z*x*y, z*x*z, z*y*x, z*y^2, z*y*z, z^2*x, z^2*y, z^3)
sage: Q.0
xbar
sage: Q.1
ybar
sage: Q.2
zbar
sage: Q.0*Q.1
xbar*ybar
sage: Q.0*Q.1*Q.0
0
"""
from sage.rings.quotient_ring import QuotientRing
return QuotientRing(self, I, names=names)