Let $R$ be a $k$-algebra of Krull dimension one where $k$ denotes a field. Let $I,J \subseteq R$ be two ideals of $R$ of dimension zero (that is $R/I$ has Krull dimension zero). Is it true that $$\dim_k R/(IJ) \leq \dim_k R/I + \dim_k R/J$$ holds in general? It is easily seen to be true if one of the factors is invertible. In this case we even have equality.
But I am wondering whether we have a statement as above?! Is there a counter-exmple? Does the statement hold with more assumptions on $I$ and $J$? For instance it looks promising for me to assume $IJ = I \cap J$.
Thank you very much for your help!
In the presence of the condition $IJ = I \cap J$, one can use the following exact sequence. $$ 0 \to R/(I \cap J) \to R/I \oplus R/J \to R/ (I+J) \to 0. $$ As each component is finite-dimensional (otherwise, the RHS is infinity), the dimension of the middle is the sum of the dimensions of the outer two).
For a counter-example, take $R = k[x,y]/ (xy)$ and $I=J = (x^2,y^2)R$. Then $R/I = k[x,y]/(x^2,xy,y^2)$ is of dimension $3$, but $R/IJ = k[x,y]/(x^4,xy,y^4)$ is of dimension 7.