Add elimination for ideals over noncommutative rings

[miguelmarco]

This ticket adds the option of eliminating variables from noncommutative ideals.

CC: @tscrim @simon-king-jena @johnperry-math

Component: algebra

Author: Miguel Marco

Branch: u/mmarco/eliminate_noncommutative_ideal

Issue created by migration from https://trac.sagemath.org/ticket/26911

[miguelmarco]

Branch: u/mmarco/eliminate_noncommutative_ideal

[miguelmarco]

Changed branch from u/mmarco/eliminate_noncommutative_ideal to none

[miguelmarco]

Author: Miguel Marco

[simon-king-jena]

comment:3

I'm puzzled. Didn't we already have this, of course in the case where Gröbnerbases of twosided ideals exist (such as G-Algebras that are implemented in Singular)?

[miguelmarco]

comment:4

I also expected this to be already available, but I wasn't able to find it, so I implemented (it is just a matter of adding a method that wraps the corresponding singular call).

[miguelmarco]

comment:5

By the way, I am tempted to implement this also for ideals over quotient rings (by lifting to the cover ring, adding the ideal that acts as kernel of the quotient, compute the elimination there and then project again). But i want to hear your thoughts on that because:

1. I am not completely sure if that mathematically makes sense. and
2. I am also not sure what should be the right class to implement it in.

So, what do you think?

[simon-king-jena]

comment:6

Replying to @miguelmarco:

By the way, I am tempted to implement this also for ideals over quotient rings (by lifting to the cover ring, adding the ideal that acts as kernel of the quotient, compute the elimination there and then project again). But i want to hear your thoughts on that because:

1. I am not completely sure if that mathematically makes sense.

I think so.

and
2) I am also not sure what should be the right class to implement it in.

So, what do you think?

I am VERY MUCH sure that it is implemented. Actually I clearly remember writing examples a couple of years ago for quotient rings of non-commutative rings. Of course, the general ideal containment problem for two-sided ideals in non-commutative rings is unsolvable. But the code contains slots for a "reduction()" method (or perhaps it was called "reduce()"). And IF that method is implemented, then quotient rings automatically are provided.

I don't have time to look into it right now, but I am sure everything can be found, for example, in sage.algebras.letterplace (it provides testing equality of cosets modulo two-sided weighted-homogeneous ideals).

[simon-king-jena]

comment:7

Ouch. Maybe I was totally misunderstanding what this ticket is about. I thought it was about "reduction modulo Gröbner bases" ("computing normal forms"). Now I realise that you talk about elimination of variables, which of course is something completely different.

So, perhaps it is needed to disregard my previous comments...

[miguelmarco]

Branch: u/mmarco/eliminate_noncommutative_ideal

[embray]

comment:9

Retarging tickets optimistically to the next milestone. If you are responsible for this ticket (either its reporter or owner) and don't believe you are likely to complete this ticket before the next release (8.7) please retarget this ticket's milestone to sage-pending or sage-wishlist.

[embray]

comment:10

Ticket retargeted after milestone closed (if you don't believe this ticket is appropriate for the Sage 8.8 release please retarget manually)

[embray]

comment:11

As the Sage-8.8 release milestone is pending, we should delete the sage-8.8 milestone for tickets that are not actively being worked on or that still require significant work to move forward. If you feel that this ticket should be included in the next Sage release at the soonest please set its milestone to the next release milestone (sage-8.9).