Trace ideals and uniserial modules

TL;DR

This paper explores the properties of trace ideals of projective modules over endomorphism rings of uniserial modules, providing new insights into their asymmetries and structural characteristics.

Contribution

It offers an intrinsic description of when trace ideals differ between left and right modules and presents an alternative approach to constructing non-serial summands.

Findings

Identifies conditions for asymmetry of trace ideals in uniserial module endomorphism rings.

Provides an intrinsic characterization of trace ideals that are not two-sided.

Introduces a new method for constructing non-serial summands using lifting theory.

Abstract

We thoroughly investigate the trace ideals of projective modules over the endomorphism ring of a uniserial module. After the work of Dubrovin and Puninski, it is known that this class of rings provides examples of trace ideals of projective right modules that are not trace ideals of projective left modules. In this paper we further investigate when this happens, giving an intrinsic description of such trace ideals and their properties. We also use the theory associated to lifting projective modules modulo a trace ideal to give an alternative approach to Puninski's construction of a direct summand of a serial module that is not serial.