On Friday, January 2, 2015 2:36:47 AM UTC-8, Jeroen Demeyer wrote: > > Currently in Sage one can do the following: > > sage: M = (ZZ^2)*(1/2) > ... > One sees that really two rings are involved: > (1) the ring ZZ which is the ring of scalars, the ring R such that we > have an R-module. > (2) the ring QQ from which the elements (the coefficients of the > vectors) come. >
The operation you apply to obtain the model isn't defined for a free module over a ring. Extra structure is required: M needs to be embedded in a module over a ring in which (1/2) exists. The fact that we have that operation, suggests that (ZZ^2) is not just an abstract free module over ZZ, but lives in a category with more structure. Indeed: sage: A=(ZZ^2) sage: M=A*(1/2) sage: B=M.ambient_module() sage: M.ambient_vector_space() #there is an attribute of a Vector space of dimension 2 over Rational Field sage: A==B True sage: A.is_submodule(M) True sage: M.is_submodule(A) False sage: parent(A.basis()[0]) Ambient free module of rank 2 over the principal ideal domain Integer Ring sage: C=parent(M.basis()[0]) sage: parent(C.basis()[0]) is C True sage: parent(A.basis()[0]) is A True sage: C Free module of degree 2 and rank 2 over Integer Ring Echelon basis matrix: [1/2 0] [ 0 1/2] sage: B.is_submodule(C) True As you can see, there is an attribute "ambient_vectorspace", so these modules M come with an embedding into M tensor_ZZ QQ, and as you can see in the examples above, for M,C these embeddings are not the standard ones. I think that the "base ring" of these modules is unambiguous: it's ZZ. It just happens that their basis might not be represented using the base ring. The distinction between B and C comes probably from the fact that B is represented entirely relative to A, whereas C is a representation more directly tied to the ambient vector space. When working with modules over a PID, it's probably fine to use this richer category. However, we should allow people to "just" work with ZZ-modules, so I'm -1 for changing the meaning of base_ring. For someone considering purely ZZ-modules it is perhaps a surprise that A*c did not produce a submodule of A, but then the scalar c used wasn't in ZZ, so in just the category of ZZ-modules, the operation isn't defined. Note that for "ambient" modules the basis always seems to lie in the ambient module. It's just that the coefficients of the basis elements need not lie in ZZ, but that's just consistent. We have: sage: parent(N.basis()[0][0]) Integer Ring sage: parent(M.basis()[0][0]) Rational Field so it's only when we constructed a module that really requires the embedding in a vector space that we see that the coefficients of the relative basis don't live in ZZ. (what is implemented is really a "fractional module" over a PID, in analogy with fractional ideals) -- You received this message because you are subscribed to the Google Groups "sage-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-devel@googlegroups.com. Visit this group at http://groups.google.com/group/sage-devel. For more options, visit https://groups.google.com/d/optout.
