A commutative ring with unity which does not have relatively pseudo-injective ideals with zero intersection

Let $R$ be a ring with $1$ and $M$ and $N$ be any right $R$-modules. We say that $M$ is pseudo-$N$-injective if every $R$-monomorphism $f:X \to M$ from a submodule $X_R$ of $N_R$ can be extended to $N$. The modules $M$ and $N$ are called relatively pseudo-injective if $M$ is pseudo-$N$-injective and $N$ is pseudo-$M$-injective.

Can we find a commutative ring $R$ with unity such that there exist two ideals $A,B \subseteq R$ such that $A \cap B=0$ and $A_R$ is NOT pseudo-$B_R$-injective.