I think it is too much to expect general PIDs to have unique (canonical) echelon forms, since that would, as a special case, mean a canonical generator for each principal ideal. Of course there are PIDs (such as Z) where there is a natural choice, but not in general.
John On Tue, Jul 5, 2011 at 10:52 PM, Rob Beezer <goo...@beezer.cotse.net> wrote: > Looks to me like there is a problem with free module equality. Note > that this seems to be for general PID's, but not ZZ, thus the simple > example over a polynomial ring. > > 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. > > 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] > > (a) Can this echelon form be made unique, or is that too much to > expect over general PIDs? > > (b) Should __cmp__ for free modules be adjusted to test submodules, > as above, when the echelon forms are different? > > It appears that it would be easy to exploit the non-uniqueness of the > echelon form to get inconsistencies in the ordering of free modules, > since presently the ordering of free modules uses the ordering of the > module's echelonized basis matrices as its final characterizing > property. > > Rob > > -- > To post to this group, send an email to sage-devel@googlegroups.com > To unsubscribe from this group, send an email to > sage-devel+unsubscr...@googlegroups.com > For more options, visit this group at > http://groups.google.com/group/sage-devel > URL: http://www.sagemath.org > -- To post to this group, send an email to sage-devel@googlegroups.com To unsubscribe from this group, send an email to sage-devel+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/sage-devel URL: http://www.sagemath.org
