Dedekind Superrings and Related Concepts

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.