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We study a notion called $n$-standardness (defined by M. E. Rossi and extended in this paper) of ideals primary to the maximal ideal in a Cohen-Macaulay local ring and some of its consequences. We further study conditions under which the maximal ideal is three-standard, first proving results when the residue field has prime characteristic and then using the method of reduction to prime characteristic to extend the results to the equicharacteristic zero case. As an application, we extend a result due to T. Puthenpurakal and show that a certain length associated to a minimal reduction of the maximal ideal does not depend on the minimal reduction chosen.
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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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