Abstract
We study the depth and the Cohen–Macaulay property of monomial ideals whose particular intersections of their associated prime ideals are associated radical ideals. In this regard, we introduce monomial ideals with complete \({\text {assrad}}\). For such an ideal I, showing \({\text {depth}}(I)={\text {size}}(I)+1\), we obtain that I satisfies Stanley’s inequality. Furthermore, we find some classes of square-free monomial ideals whose large enough symbolic powers have complete \({\text {assrad}}\). For these ideals, we derive an upper bound for the index of depth stability of symbolic powers which is significantly smaller than the general bound given by Hoa et al. (J Algebra 473:307–323, 2017).
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Bandari, S., Jafari, R. Minimal depth of monomial ideals via associated radical ideals. Arch. Math. 117, 495–507 (2021). https://doi.org/10.1007/s00013-021-01656-3
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DOI: https://doi.org/10.1007/s00013-021-01656-3
Keywords
- Associated radical ideals
- Complete \({\text {assrad}}\)
- Depth
- Minimal depth
- Size of monomial ideals
- Stanley depth