Basic implementation of one- and twosided ideals of non-commutative rings, and quotients by twosided ideals

[simon-king-jena]

It was suggested that my patch for #7797 be split into several parts.

The first part shall be about ideals in non-commutative rings. Aim, for example:

sage: A = SteenrodAlgebra(2)
sage: A*[A.0,A.1^2]
Left Ideal (Sq(1), Sq(1,1)) of mod 2 Steenrod algebra
sage: [A.0,A.1^2]*A
Right Ideal (Sq(1), Sq(1,1)) of mod 2 Steenrod algebra
sage: A*[A.0,A.1^2]*A
Twosided Ideal (Sq(1), Sq(1,1)) of mod 2 Steenrod algebra

It was suggested to also add quotients by twosided ideals, although it will be difficult to provide examples before having letterplace ideals.

Apply:

• attachment: trac11068_nc_ideals_and_quotients.patch
• attachment: trac11068_quotient_ring_without_names.patch
• attachment: trac11068_lifting_map.patch

Depends on #9138
Depends on #11900
Depends on #11115

CC: @nthiery @jhpalmieri

Component: algebra

Keywords: onesided twosided ideal noncommutative ring sd32

Author: Simon King

Reviewer: John Perry

Merged: sage-5.0.beta1

Issue created by migration from https://trac.sagemath.org/ticket/11068

[simon-king-jena]

Description changed:

--- 
+++ 
@@ -11,3 +11,5 @@
 sage: A*[A.0,A.1^2]*A
 Twosided Ideal (Sq(1), Sq(1,1)) of mod 2 Steenrod algebra
 ```
+
+It was suggested to also add quotients by twosided ideals, although it will be difficult to provide examples before having letterplace ideals.

[simon-king-jena]

Work Issues: Add examples; move code from ring.py to rings.py

[simon-king-jena]

Description changed:

--- 
+++ 
@@ -13,3 +13,5 @@
 ```
 
 It was suggested to also add quotients by twosided ideals, although it will be difficult to provide examples before having letterplace ideals.
+
+Depends on #10961

[simon-king-jena]

comment:2

Depends on #10961

The patch is incomplete, since proper examples are missing. Proper examples will be provided by #7797, but Nicolas promised to try and find some "small" example.

Really, all what we need is a non-commutative ring and something that inherits from Ideal_nc and provides a reduce() method yielding normal forms.

Also, that patch is preliminary, since it copies some code from sage/rings/ring.pyx to sage/categories/rings.py, rather than moving it.

[simon-king-jena]

comment:3

Concerning examples.

I suggest to make a standard example out of the following emulation of a certain very simple ideals of a free algebra, without letterplace:

sage: from sage.rings.noncommutative_ideals import Ideal_nc
# mainly we need a 'reduce()' method. So, provide one!
sage: class PowerIdeal(Ideal_nc):
....:     def __init__(self, R, n):
....:         self._power = n
....:         Ideal_nc.__init__(self,R,[x^n for x in R.gens()])
....:     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: F.<x,y,z> = FreeAlgebra(QQ, 3)
sage: I3 = PowerIdeal(F,3); I3
Twosided Ideal (x^3, y^3, z^3) of Free Algebra on 3 generators (x, y, z) over Rational Field
# We need to use super in order to access the generic quotient:
sage: Q3.<a,b,c> = super(F.__class__,F).quotient(I3)
sage: Q3
Quotient of Free Algebra on 3 generators (x, y, z) over Rational Field by the ideal (x^3, y^3, z^3)
sage: (a+b+2)^4
16 + 32*x + 32*y + 24*x^2 + 24*x*y + 24*y*x + 24*y^2
sage: Q3.is_commutative()
False
sage: I2 = PowerIdeal(F,2); I2
Twosided Ideal (x^2, y^2, z^2) of Free Algebra on 3 generators (x, y, z) over Rational Field
sage: Q2.<a,b,c> = super(F.__class__,F).quotient(I2)
sage: Q2.is_commutative()
True
sage: (a+b+2)^4
16 + 32*x + 32*y

I think I would be able to add this example as doc test tomorrow, also removing the whole ideal and quotient stuff from sage.rings.ring.Ring to sage.categories.rings.Rings.ParentMethods.

[nthiery]

comment:4

Nice! That sounds like a perfect example indeed.

Thanks for removing this item from my TODO list :-)

[simon-king-jena]

comment:5

Replying to @simon-king-jena:

...
I think I would be able to add this example as doc test tomorrow, also removing the whole ideal and quotient stuff from sage.rings.ring.Ring to sage.categories.rings.Rings.ParentMethods.

Apparently it took a lot longer. The problem is "moving the whole ideal and quotient stuff from sage.rings.ring.Ring to sage.categories.rings.Rings.ParentMethods. Namely, that involves a couple of methods with cached output (the cache being hand-written). One could use the @cached_method decorator in the parent methods -- but the problem is that the cache breaks when one has a ring that does not allow attribute assignment.

That problem is solved at #11115. Moreover, @cached_method is now faster than a hand-written cache in Python. However, it is slower than a hand-written cache in Cython.

So, the question is: Can we accept the slow-down that would result from moving code from the ring class to the ring category? Or is it easier to accept some duplication of code?

[simon-king-jena]

comment:6

Replying to @simon-king-jena:

So, the question is: Can we accept the slow-down that would result from moving code from the ring class to the ring category? Or is it easier to accept some duplication of code?

Any answer? Probably one would first need to see how much of a slow-down we will really find.