A Unified Approach to the Decomposition Theorems in Baer $$*$$-Rings

The aim of the paper is to generalize decomposition theorems showed in Bagheri-Bardi et al. (Linear Algebra Appl 583:102–118, 2019; Linear Algebra Appl 539:117–133, 2018) by a unified approach. We show a general decomposition theorem with respect to a hereditary property. Then the vast majority of decompositions known in the algebra of Hilbert space operators is generalized to elements of Baer $$*$$ ∗ -rings by this theorem. The theorem yields also results which are new in the algebra of bounded Hilbert space operators. Additionally, the model of summands in Wold–Słociński decomposition is given in Baer $$*$$ ∗ -rings.

Skew Hurwitz serieswise quasi-armendariz rings

For a ring endomorphism [Formula: see text], a generalization of semiprime rings and right p.q.-Baer rings, which we call quasi-Armendariz rings of skew Hurwitz series type (or simply, [Formula: see text]-[Formula: see text]), is introduced and studied. It is shown that the [Formula: see text]-rings are closed upper triangular matrix rings, full matrix rings and Morita invariance. Some characterizations for the skew Hurwitz series ring [Formula: see text] to be quasi-Baer, generalized quasi-Baer, primary, nilary, reflexive, ideal-symmetric and semiprime are concluded.

Skew inverse Laurent series extensions of weakly principally quasi Baer rings

A ring [Formula: see text] is called weakly principally quasi Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is left [Formula: see text]-unital by left semicentral idempotents, which implies that [Formula: see text] modulo the right annihilator of any principal right ideal is flat. We study the relationship between the weakly p.q.-Baer property of a ring [Formula: see text] and those of the skew inverse series rings [Formula: see text] and [Formula: see text], for any automorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text] of [Formula: see text]. Examples to illustrate and delimit the theory are provided.

The reduced ring order and lower semi-lattices II

This paper continues the study of the reduced ring order (rr-order) in reduced rings where [Formula: see text] if [Formula: see text]. A reduced ring is called rr-good if it is a lower semi-lattice in the order. Examples include weakly Baer rings (wB or PP-rings) but many more. Localizations are examined relating to this order as well as the Pierce sheaf. Liftings of rr-orthogonal sets over surjections of reduced rings are studied. A known result about commutative power series rings over wB rings is extended, via methods developed here, to very general, not necessarily commutative, power series rings defined by an ordered monoid, showing that they are wB.

Partial orders on the power sets of Baer rings

Let [Formula: see text] be a ring. Motivated by a generalization of a well-known minus partial order to Rickart rings, we introduce a new relation on the power set [Formula: see text] of [Formula: see text] and show that this relation, which we call “the minus order on [Formula: see text]”, is a partial order when [Formula: see text] is a Baer ring. We similarly introduce and study properties of the star, the left-star, and the right-star partial orders on the power sets of Baer ∗-rings. We show that some ideals generated by projections of a von Neumann regular and Baer ∗-ring [Formula: see text] form a lattice with respect to the star partial order on [Formula: see text]. As a particular case, we present characterizations of these orders on the power set of [Formula: see text], the algebra of all bounded linear operators on a Hilbert space [Formula: see text].