Structure of modules stably annihilated by a fixed ideal

Let $R$ be a commutative noetherian ring, and denote by $\mathrm{mod}R$ the category of finitely generated $R$-modules. In this paper, for an ideal $I$ of $R$, we introduce the full subcategory ${\mathrm{mod}}_I(R)$ of $\mathrm{mod}R$ consisting of modules whose stable annihilators contain $I$, and we investigate its structure. As an application, we explore the syzygy category of maximal Cohen–Macaulay $R$-modules, extending a theorem of Dey and Liu from the Gorenstein case to the Cohen–Macaulay case.