1. Tensor Products of R-modules

If I'm working with R-module tensors, is it true that

(s1 x r1)(s2 x r2)= (s1s2 x r1r2), (where x should have a circle around it)?

The question comes from this:
Establish an isomorphism between two R-algebras « Project Crazy Project .

Also, why are we sure 1 is in R? (It looks like they're assuming this is true, b/c to my knowledge R[x] is the set of polynomials with coefficients from R, but they have (1x1) at one point, and (s x x^i). )

2. Re: Tensor Products of R-modules

Originally Posted by gummy_ratz
If I'm working with R-module tensors, is it true that

(s1 x r1)(s2 x r2)= (s1s2 x r1r2), (where x should have a circle around it)?

The question comes from this:
Establish an isomorphism between two R-algebras « Project Crazy Project .

Also, why are we sure 1 is in R? (It looks like they're assuming this is true, b/c to my knowledge R[x] is the set of polynomials with coefficients from R, but they have (1x1) at one point, and (s x x^i). )
This is how we naturally define the ring operation on the tensor product of algebras, so yes. And yes, $\displaystyle R[x]$ is unital, and we are assuming that that $\displaystyle R$ is a unital subring.