Unifying ideal and radical theory: a framework of S-structures

Abstract

The foundational theorems of commutative algebra are often predicated on the absence of zero divisors. This paper systematically removes this constraint by developing a comprehensive theory of rings relative to a multiplicatively closed set S. We establish S-maximal ideals as the central tool in a coherent framework that also includes S-fields and S-local rings. A cornerstone of our theory is a structural characterization of S-local rings that we prove these are precisely the rings whose non-S-unit elements form an ideal. This result is built upon an S-analogue of Krull's Theorem, which characterizes the newly introduced S-Jacobson radical entirely in terms of S-units. Furthermore, we resolve the asymmetry between S-prime and S-maximal ideals by demonstrating their equivalence for significant classes of rings, including S-Boolean and S-von Neumann regular rings. Collectively, our findings provide a robust ideal-theoretic framework for analyzing the structure of rings that are not necessarily integral domains.