Invertibility in nonassociative ordered rings and in weak-quasi-topological nonassociative rings

Abstract

Invertibility is important in ring theory because it enables division and facilitates solving equations. Moreover, rings can be endowed with extra ''structure'' such as order and topology that add new properties. The two main theorems of this article are contributions to invertibility in the context of ordered and weak-quasi-topological rings. Specifically, the first theorem asserts that the interval $]0,1]$ in any suitable partially ordered ring consists entirely of invertible elements. The second theorem asserts that if $f$ is a norm from a ring to a partially ordered ring endowed with interval topology, then under certain conditions, the subset of elements such that $f(1-a) < 1$ consists entirely of invertible elements. The second theorem relies on the assumption of sequential Cauchy completeness of the topology induced by the norm $f$, which as we recall, takes values in an ordered ring endowed with the interval topology (an example of a coarse topology). The fact that a ring endowed with the topology associated with a seminorm into an ordered ring endowed with the interval topology is a locally convex quasi-topological group with an additional continuity property of the product is dealt with in a separate section. A brief application to frame theory is also included.