Free module equality, comparisons

[rbeezer]

On #11553, mmarco gives an example that can be distilled to:

sage: R = PolynomialRing(QQ, 'a')
sage: x = vector(R, [1, 0])
sage: y = vector(R, [0, 1])
sage: z = vector(R, [0,-1])
sage: A = (R^2).span([x, y])
sage: B = (R^2).span([x, z])
sage: A == B
False
sage: A.is_submodule(B)
True
sage: B.is_submodule(A)
True

Root cause looks like an assumption that echelon form (Hermite form) over PIDs is unique, which may be insurmountable in general.

sage: S = matrix([x, y])
sage: S._echelon_form_PID()[1]
[1 0]
[0 1]
sage: T = matrix([x, z])
sage: T._echelon_form_PID()[1]
[ 1  0]
[ 0 -1]

This also affects the ordering of free modules via the __cmp__ method.

Informative discussion at:
http://groups.google.com/group/sage-devel/browse_thread/thread/eda6f3ebc118b8b

CC: @sagetrac-fwclarke

Component: linear algebra

Stopgaps: todo

Issue created by migration from https://trac.sagemath.org/ticket/11579

[rbeezer]

comment:1

One suggestion for a fix:

(a) make comparisons raise a NotImplementedError for rings that are not fields and not ZZ - though I have not checked to see how important an ordering on free modules is.

(b) use a pair of calls to .is_submodule() to test equality.

I don't plan to pursue this one much further, but could review a fix.

[sagetrac-jakobkroeker]

Stopgaps: todo

[simonbrandhorst]

comment:9

Be aware of #23978 Rich comparison for Modules

[simonbrandhorst]

comment:10

sage: sage: R = PolynomialRing(QQ, 'a')
....: sage: x = vector(R, [1, 0])
....: sage: y = vector(R, [0, 1])
....: sage: z = vector(R, [0,-1])
....: sage: A = (R^2).span([x, y])
....: sage: B = (R^2).span([x, z])
....: sage: A == B
....: 
True
sage: A.is_submodule(B)
True
sage: B.is_submodule(A)
True

Looks fixed to me.
Close?

[fchapoton]

comment:13

let us close

[videlec]

comment:14

closing positively reviewed duplicates