Linear recurrent sequences over commutative rings

For a Noetherian commutative ring $\mathbf R$ with a unity there exist Galois correspondences between the structure of finitely generated submodules of the $R[x]$-module $\mathcal L_\mathbf R$ of all linear recurrent sequences (LRS) over $R$ and the structure of unitary ideals (the annihilators of these modules) in $R[x]$. We prove that these correspondences are one-to-one if and only if $R$ is a quasi-Frobenius ring. In this case we show that the well-known relations between sums and intersections of modules and their annihilators for LRS over fields are preserved. In the case when $R$ is also a principal ideal ring we construct a system of generators for the module of all LRS that are annihilated by a given unitary ideal, and derive a test for the cyclicity of this module over the ring $R[x]$.