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Attached primes under skew polynomial extensions

机译：偏多项式扩展下的附加质数

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摘要

In the author's work [S. A. Annin, Attached primes over noncommutative rings, J. Pure Appl. Algebra 212 (2008) 510521], a theory of attached prime ideals in noncommutative rings was developed as a natural generalization of the classical notions of attached primes and secondary representations that were first introduced in 1973 as a dual theory to the associated primes and primary decomposition in commutative algebra (see [I. G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math. 11 (1973) 2343]). Associated primes over noncommutative rings have been thoroughly studied and developed for a variety of applications, including skew polynomial rings: see [S. A. Annin, Associated primes over skew polynomial rings, Commun. Algebra 30(5) (2002) 25112528; and S. A. Annin, Associated primes over Ore extension rings, J. Algebra Appl. 3(2) (2004) 193205]. Motivated by this background, the present article addresses the behavior of the attached prime ideals of inverse polynomial modules over skew polynomial rings. The goal is to determine the attached primes of an inverse polynomial module M[x-1] over a skew polynomial ring R[x;σ] in terms of the attached primes of the base module MR. This study was completed in the commutative setting for the class of representable modules in [L. Melkersson, Content and inverse polynomials on artinian modules, Commun. Algebra 26(4) (1998) 11411145], and the generalization to noncommutative rings turns out to be quite non-trivial in that one must either work with a Bass module MR or a right perfect ring R in order to achieve the desired statement even when no twist is present in the polynomial ring "Let MR be a module over any ring R. If M[x -1]R is a completely σ-compatible Bass module, then Att(M[x-1]S) = {p[x]:p ∈ Att(MR)}." The sharpness of the results are illustrated through the use of several illuminating examples.
机译：在作者的作品中[S. A.安宁，非交换环上的素数，J。Pure Appl。 [Algebra 212（2008）510521]，作为对交换素数和次要表示形式的经典概念的自然概括，发展了一种非交换环上的附加素理想的理论，该概念于1973年作为对相关素数和原始分解的对偶论首次引入在交换代数中（请参阅[IG Macdonald，交换环上模块的次级表示，Sympos。Math。11（1973）2343]）。非交换环上的相关质数已经被广泛研究和开发用于多种应用，包括偏多项式环： A. Annin，偏多项式环上的相关质数，Commun。代数30（5）（2002）25112528; S. A. Annin，《矿石延伸环上的相关素数》，J。Algebra Appl。 3（2）（2004）193205]。基于这种背景，本文探讨了偏多项式环上逆多项式模块的附加素理想的行为。目的是根据基本模块MR的附加质数确定偏多项式环R [x;σ]上的逆多项式模块M [x-1]的附加质数。这项研究是在可交换的环境中完成的。 Melkersson，Artinian模块上的内容和逆多项式，Commun。 Algebra 26（4）（1998）11411145]，而对非交换环的推广却是不平凡的，因为必须使用Bass模块MR或正确的理想环R才能获得所需的语句，甚至当多项式环中没有扭曲时，“让MR成为任何环R上的模块。如果M [x -1] R是完全σ兼容的Bass模块，则Att（M [x-1] S）= { p [x]：p∈Att（MR）}。”通过使用几个示例说明了结果的清晰度。

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