I have the following problem:
Let $R$ be a ring and $a,b\in R$ ideals in $R$. Show: $$R/a\otimes_\mathbb{R} R/b\rightarrow R/(a\cup b)$$ $$[x]\otimes[y]\rightarrow[x*y]$$ is a welldefined $R$-modulisomorphism.
I have two ideas:
$1.$ I could construct an inverse Isomorphism.
$2.$ I could use the compatibility of tensorproducts and quotients.
The problem is I found no solution using either way. Can someone help me please?