Algebraic theory of Colombeauʼs generalized numbers

The ring of Colombeau’s full generalized quaternions

“…The Proposition 2.6 below appears in [5] and we will extend it for H in Proposition 3.11 in the end Section 3. Proposition 2.6 (Convexity of ideals).…”

“…where D is the ultra-metric in K defined in [5] inspired by Scarpalezos for the Colombeau's simplifies generalized numbers, K s . Denote by Inv(K) the unit group of K. Let…”

“…In [5] it is proved that g f (F p ) is indeed a minimal prime ideal of K. In general g f (F p ) is not closed and so K is not Von Neumann regular.…”

“…for any x ∈ K. Thus, we have that an element x ∈ K is a unit iff there exists r ∈ R, such that |x| ≥α r (see Theorem 2.2). In [5], it is also proved that for 0 < x ∈ K, there exists y ∈ K, such that x = y 2 . In Sections 3 and 4, we will use freely some of the results in this section.…”

“…The notation used is mostly standard. Some important references for the theory of Colombeau's full generalized numbers, functions and their topologies are in [13,2,10,15,20,22], and more recently, see [5,25] and [11]. See also the set of Notation 1.1 of [5], for example, items a), i), j), l) and m) to understand some notations which appear throughout this paper.…”