Let R be a commutative ring with unity. Given two algebras A and A' over a field K, both with R-grading form. Prove that B:= A \otimes_R A' (the tensor product of A and A') has a R²-grading.
Any tip of how to prove this?
Dear Kelvin,
I think you meant: B=A\otimes _K A' (i.e. tensor product over the field K). Next, we should prove this by definition: if a\in A and deg(a)=r\in R, a'\in A' and deg(a')=r'\in R are two R-homogeneous elements then we assign deg(a\otimes a')=(r,r')\in R2. It remains to show that this grading is compatible with multiplication in B=A\otimes _K A' (which is not hard).
Best regards, Yuri
Dear Kelvin,
I think you meant: B=A\otimes _K A' (i.e. tensor product over the field K). Next, we should prove this by definition: if a\in A and deg(a)=r\in R, a'\in A' and deg(a')=r'\in R are two R-homogeneous elements then we assign deg(a\otimes a')=(r,r')\in R2. It remains to show that this grading is compatible with multiplication in B=A\otimes _K A' (which is not hard).
Best regards, Yuri
Thank you Yuri, your contributions are always helping me. And helping this network.
Sure I committed a mistake writing R instead of K when defined B.
I was only thinking about the allowance of assigning a degree to a b \in B when the R²-graduation of B is my thesis. But now I get it, this is enough once I prove the compatibility with the operation in B.
Thank you once again.
- Badges
- Science topic
- Similar topics
- Mathematical Sciences
- Algebra