Abstract

We give a full characterization of the unique Ratliff–Rush complete ideals in the polynomial rings R with two variables over a field. For a class of monomial ideals in R, we investigate the Ratliff–Rush behavior of powers of these ideals and also the depth of their associated graded ring. Namely, we give some sufficient and some necessary conditions for this depth to be positive. The results of this paper allow us to answer several questions of [13 Heinzer, W., Johnston, B., Lantz, D., Shah, K. (1993). Coefficient ideals in and blowups of a commutative Noetherian domain. J. Algebra 162(2):355–391. DOI: 10.1006/jabr.1993.1261.[Crossref], [Web of Science ®] , [Google Scholar]].

摘要

我们给出了多项式环R 中唯一的 Ratliff-Rush 完全理想的完整特征，在一个域上具有两个变量。对于R 中的一类单项式理想，我们研究了这些理想的幂的 Ratliff-Rush 行为以及它们相关分级环的深度。也就是说，我们给出了这个深度为正的一些充分条件和一些必要条件。本文的结果使我们能够回答[ 13 Heinzer, W.、Johnston, B.、Lantz, D.、Shah, K. ( 1993 )。可交换 Noetherian 域中的系数理想和爆炸。J. 代数162(2): 355 – 391。DOI：10.1006/jabr.1993.1261。[Crossref]、[Web of Science®]、[  Google Scholar] ]。