Identifying elements in V(x)W, V,W V.Spaces

[WWGD]

Hi, everyone:
I have been playing around with tensor products of vector spaces recently.
This question came up:

How can we tell if c(a(x)b)~(a(x)b) , or wether any two general elements
are in the same class, other than in the std. cases like c(a(x)b)~(ca(x)v)~
(a(x)cv) , etc.

Any Ideas?.
Thanks.

 

[mathwonk]

this is why tensor products are hard, it is hard to tell when two reporesentatives for elements represent the same element of a tensor product.

as a tiny example, show that every pair of representatives in the tensor product of Z/(37) and Z/(101) represent the same element. e.g. for all pairs of integers (n,m) and (a,b), their tensor products are equal in that group.

 

[WWGD]

Thanks, Mathwonk. I hope I am not missing something obvious
here, but I am kind of confused, tho; are you seeing Z/(37) and
Z/(101) as fields?. How do you define the tensor product, then?

 

[WWGD]

Sorry for my dumbness here, Mathwonk. I was stuck with the idea of tensoring
vector spaces, and was thinking of both Z/(37) and Z/(101) as vector
spaces over themselves , so that the tensor did not make sense. Then
I thought for a second and realized it is a Z-module tensor.I'll do the exercise--
and review my algebra.

 

[mathwonk]

the case of vector spaces is trivial. the dimension of VtensW is always the product of their dimensions, and if {vi} and {wj} are bases, then so is {vitenswj}.
so every element can be written in terms of the basis to see if two elements are equivalent. the hard cases are modules that are not free.