the docstring for the associated_primes method on multivariate polynomial ideals is wrong #7959
Description
The docstring for associated_primes claims it returns a list of pairs (I,P), but in fact it just returns the P. So this is wrong.
@require_field
@redSB
def associated_primes(self, algorithm='sy'):
r"""
Return a list of primary ideals (and their associated primes) such
that their intersection is `I` = ``self``.
An ideal `Q` is called primary if it is a proper ideal of
the ring `R` and if whenever `ab \in Q` and
`a \not\in Q` then `b^n \in Q` for some
`n \in \ZZ`.
If `Q` is a primary ideal of the ring `R`, then the
radical ideal `P` of `Q`, i.e.
`P = \{a \in R, a^n \in Q\}` for some
`n \in \ZZ`, is called the
*associated prime* of `Q`.
If `I` is a proper ideal of the ring `R` then there
exists a decomposition in primary ideals `Q_i` such that
- their intersection is `I`
- none of the `Q_i` contains the intersection of the
rest, and
- the associated prime ideals of `Q_i` are pairwise
different.
This method returns the associated primes of the `Q_i`.
INPUT:
- ``algorithm`` - string:
- ``'sy'`` - (default) use the shimoyama-yokoyama algorithm
- ``'gtz'`` - use the gianni-trager-zacharias algorithm
OUTPUT:
- ``list`` - a list of primary ideals and their
associated primes [(primary ideal, associated prime), ...]
EXAMPLES::
sage: R.<x,y,z> = PolynomialRing(QQ, 3, order='lex')
sage: p = z^2 + 1; q = z^3 + 2
sage: I = (p*q^2, y-z^2)*R
sage: pd = I.associated_primes(); pd
[Ideal (z^3 + 2, y - z^2) of Multivariate Polynomial Ring in x, y, z over Rational Field,
Ideal (z^2 + 1, y + 1) of Multivariate Polynomial Ring in x, y, z over Rational Field]
Component: commutative algebra
Author: Willem Jan Palenstijn
Reviewer: William Stein
Merged: sage-4.3.1.rc1
Issue created by migration from https://trac.sagemath.org/ticket/7959