Dedekind Superrings and Related Concepts

TL;DR

This paper explores Dedekind superrings within the $\

Contribution

It develops the theory of Dedekind superrings and related modules intrinsically in the supercommutative setting, highlighting key differences from classical commutative algebra.

Findings

Classical Dedekind domain properties often fail in the superring context.

Structural parallels to classical theory are identified and contrasted.

Fundamental discrepancies are characterized in the $\

Abstract

This article investigates the properties of Dedekind superrings, invertible supermodules and projective supermodules within the $\mathbb{Z}_2$-graded framework. Rather than treating these entities as specialized instances of general noncommutative ring theory, we develop them intrinsically within the category of supercommutative superrings. We examine the structural parallels to the classical commutative framework and, more importantly, characterize the fundamental discrepancies that emerge in the $\mathbb{Z}_2$-graded setting. In particular, we show that many hallmark equivalences of classical Dedekind domains-including those involving integral closedness and the coincidence of principal and unique factorization domains-fail to persist in the presence of an odd part.