Invertibility in partially ordered nonassociative rings
Nizar El Idrissi, Hicham Zoubeir
TL;DR
The paper investigates invertibility in partially ordered nonassociative rings equipped with Frink's interval topology and related seminorm-induced topologies. It develops an oriented geometric-series framework proving that every $x$ in $]0,1]$ is invertible under suitable monotone $ leftarrow o$-completeness assumptions, with explicit inverse given by a convergent series. A second main result shows that, in the setting of a seminorm-induced topology into an ordered nonassociative ring endowed with Frink's interval topology and under Hausdorff sequential Cauchy-completeness, elements with $f(1-x)<1$ are invertible, with the inverse again given by a geometric-type sum; left-inverse results hold under stronger hypotheses. The work also establishes foundational properties of the induced topologies, including weak-quasi-topological group structures and separate continuity of multiplication, broadening the toolkit for invertibility questions in generalized ordered rings.
Abstract
Invertibility is important in ring theory because it enables division and facilitates solving equations. Moreover, (nonassociative) rings can be endowed with an extra ''structure'' such as order and topology allowing more richness in the theory. The two main theorems of this article are contributions to invertibility in the context of partially ordered nonassociative rings \textit{and} Hausdorff sequentially Cauchy-complete weak-quasi-topological nonassociative rings. Specifically, the first theorem asserts that the interval $]0,1]$ in any suitable partially ordered nonassociative ring consists entirely of invertible elements. The second theorem asserts that if $f$ is a suitably generalized concept of seminorm from a nonassociative ring to a partially ordered nonassociative ring endowed with Frink's interval topology, then under certain conditions, the subset of elements such that $f(1-a) < 1$ consists entirely of invertible elements. Part of the assumption of the second theorem is that of Hausdorff sequential Cauchy-completeness of the first ring under the topology induced by the seminorm $f$ (which takes values in a partially ordered nonassociative ring endowed with Frink's interval topology). Frink's interval topology is an example of a coarse locally-convex $T_1$ topology. Moreover, to our knowledge, the topology induced by a seminorm into a partially ordered nonassociative ring has never been introduced. Some additional original facts, such as the fact that the topology on a nonassociative ring $R_1$ induced by a norm into a totally ordered associative division ring $R_2$ endowed with Frink's interval topology (or equivalently, with the order topology, since the order of $R_2$ is total) is a Hausdorff locally convex quasi-topological group with an additional separate continuity property of the product, are dealt with in the second section ''Preliminaries''.