On the Graded Annihilators of Right Modules Over The Frobenius Skew Polynomial Ring

TL;DR

This paper investigates the structure of graded annihilators of modules over Frobenius skew polynomial rings, proving finiteness under certain conditions and providing counterexamples otherwise.

Contribution

It confirms finiteness of graded annihilators for semi-local rings and presents counterexamples in the general case, advancing understanding of module structure over Frobenius skew polynomial rings.

Findings

Finiteness of graded annihilators for semi-local rings

Counterexamples in the general case

Extension of previous theoretical results

Abstract

Let R be a commutative Noetherian ring of prime characteristic and M be an x-divisible right R[x,f]-module that is Noetherian as R-module. We give an affirmative answer to the question of Sharp and Yoshino in the case where R is semi-local and prove that the set of graded annihilators of R[x,f]-homomorphic images of M is finite. We also give a counterexample in the general case.