Tensor product of a module and a quotient module

Question

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.

The thing is, $\langle 0\otimes l\rangle_{l\in L}=0$. Perhaps you mean to quotient by $M\otimes_R L$? — Captain Lama, Aug 05 '22 at 21:22
I think it will be easier to just construct a bilinear map $R/I\times R/J\to R/(I+J)$. — wormram, Aug 05 '22 at 21:27
@Michael I wish to establish a general fact about quotients, if it is possible — Mitya Kustov, Aug 05 '22 at 21:29

score 1 · Accepted Answer · answered Aug 05 '22 at 21:46

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.

Tensor product of a module and a quotient module

1 Answers1