ATTACHED PRIMES UNDER SKEW POLYNOMIAL EXTENSIONS

Abstract

In the author's work [S. A. Annin, Attached primes over noncommutative rings, J. Pure Appl. Algebra212 (2008) 510–521], 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) 23–43]). 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. Algebra30(5) (2002) 2511–2528; and S. A. Annin, Associated primes over Ore extension rings, J. Algebra Appl.3(2) (2004) 193–205]. 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. Algebra26(4) (1998) 1141–1145], 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.