I am trying to prove the following:
Let $R$ be a commutative ring with identity and $I,J\triangleleft R$. Show that $$R/I\otimes_R R/J\simeq R/(I+J).$$
I tried to prove the following hypothesis:
Let $M,N$ be $R$-modules and let $L\subset N$ be a submodule. Then the following holds: $$M\otimes_R (N/L)\simeq \frac{M\otimes_R N}{\langle m\otimes l\rangle_{l\in L, m\in M}}.$$
My approach is to construct $R$-bilinear mapping $$B:M\times N\to M\otimes_R (N/L)$$ defined as $$B(m,n):=m\otimes [n]_L.$$ Then it seems useful to use the tensor product universal property and the first isomorphism theorem (how to check surjectivity?)
But I am even not sure about the correctness of the hypotgesis.
UPD.
Best Answer
Your hypothesis is correct, but unfortunately it is not a completely trivial thing. The technical term is to say that tensoring with $M$ is a "right exact" functor. Proving this directly is actually a little tricky, usually one would use the fact that it is a left adjoint. The accepted answer to this question also gives a more elementary way of showing that.