Here is my question:

Does there exist an infinite commutative ring $R$ with identity with an indecomposable injective (unitary) $R$-module $M$ of larger cardinality than $R$ with the additional property that $M$ has a minimum submodule (that is, a nonzero $R$-submodule $N$ such that $N\leq K$ for every nonzero $R$-submodule $K$ of $M$)? This question is equivalent to the question obtained by dropping the adjective "indecomposable injective."

This is a bit out of my research area, but I would really like to know the answer for a paper I'm working on. Any references would be much appreciated.

Thanks!