tensor product (commutative algebra)

Last updated January 27, 2022

Of algebras

1. Let $A,B$ be $R$-algebras, and $I,J$ ideals in $A,B$ respectively. Then $$(A/I)\otimes_R (B/J)\cong (A\otimes_R B)/(I\otimes_R 1+1\otimes_R J).$$
2. Let $M$ be an $A$-module. Then $$A \otimes_A M\cong M .$$