Let $(R,frak m)$ be a commutative noetherian local ring. In this paper, we prove that if $frak m$ is decomposable, then for any finitely generated $R$-module $M$ of infinite projective dimension $frak m$ is a direct summand of (a direct sum of) syzygies of $M$. Applying this result to the case where $frak m$ is quasi-decomposable, we obtain several classfications of subcategories, including a complete classification of the thick subcategories of the singularity category of $R$.
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