Non-square matrices should inherit from ModuleElement, not RingElement

[pjbruin]

Matrices inherit from RingElement even if they are not square. Their parents do behave correctly.

sage: A = Matrix([[1, 2], [3, 4]])
sage: A.parent() in Rings()
True
sage: A.parent() in Modules(ZZ)
True
sage: isinstance(A, RingElement)
True

sage: B = Matrix([[1, 2, 3], [4, 5, 6]])
sage: B.parent() in Rings()
False
sage: A.parent() in Modules(ZZ)
True
sage: isinstance(B, RingElement)
True

The same happens with Algebras(ZZ) and AlgebraElement.

Square matrices over a ring R should be made to inherit dynamically from AlgebraElement; non-square matrices should only inherit from ModuleElement.

Motivation: Multiplication of matrices over a commutative ring R is in general only a bilinear map between R-modules (in general three different ones). Only if the matrices are square of the same size n can this be described as the multiplication map on an R-algebra, namely Mat_n(R).

This is a duplicate of #804.

CC: @robertwb

Component: categories

Keywords: inheritance

Reviewer: Travis Scrimshaw

Issue created by migration from https://trac.sagemath.org/ticket/15215

[pjbruin]

Description changed:

--- 
+++ 
@@ -23,3 +23,5 @@
 
 *Motivation:* Multiplication of matrices over a commutative ring *R* is in general only a bilinear map between *R*-modules (in general three different ones).  Only if the matrices are square of the same size *n* can this be described as the multiplication map on an *R*-algebra, namely Mat<sub>*n*</sub>(*R*).
 
+This is a duplicate of #804.
+

[pjbruin]

comment:2

I propose to close this as a duplicate of #804.

[tscrim]

Reviewer: Travis Scrimshaw