Rings whose von Neumann regular elements are fine

Let $S$ be a subsemigroup which contains 0 of a torsion-free abelian (additive) group. Then $S$ is called a grading monoid (or a $g$-monoid). The group $ \{s-s'|s,s'\in S\}$ is called the quotient group of $S$, and is denored by $q(S)$. Let $R$ be a commutative ring. The total quotient ring of $R$ is denoted by $q(R)$. Throught the paper, we assume that a $g$-monoid properly contains $ \{0\}$. A commutative ring is called a ring, and a non-zero-divisor of a ring is called a regular element of the ring. We consider integral elements over the semigroup ring $ R[X;S]$ of $S$ over $R$. Let $S$ be a $g$-monoid with quotient group $G$. If $ n\alpha\in S$ for an element $ \alpha$ of $G$ and a natural number $n$ implies $ \alpha\in S$, then $S$ is called an integrally closed semigroup. We know the following fact: ${\bf Theorem~1}$ ([G2, Corollary 12.11]). Let $D$ be an integral domain and $S$ a $g$-monoid. Then $D[X;S]$ is integrally closed if and only if $D$ is an integrally closed domain and $S$ is an integrally closed semigroup. Let $R$ be a ring. In this paper, we show that conditions for $R[X;S]$ to be integrally closed reduce to conditions for the polynomial ring of an indeterminate over a reduced total quotient ring to be integrally closed (Theorem 15). Clearly the quotient field of an integral domain is a von Neumann regular ring. Assume that $q(R)$ is a von Neumann regular ring. We show that $R[X;S]$ is integrally closed if and only if $R$ is integrally closed and $S$ is integrally closed (Theorem 20). Let $G$ be a $g$-monoid which is a group. If $R$ is a subring of the ring $T$ which is integrally closed in $T$, we show that $R[X;G]$ is integrally closed in $T[X;S]$ (Theorem 13). Finally, let $S$ be sub-$g$-monoid of a totally ordered abelian group. Let $R$ be a subring of the ring $T$ which is integrally closed in $T$. If $g$ and $h$ are elements of $T[X;S]$ with $h$ monic and $gh\in R[X;S]$, we show that $g\in R[X;S]$ (Theorem 24).

We show that a cellular automaton on a mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from [S.-Törmä, 2015] that von Neumann regularity is decidable condition, and we decide it for all elementary CA.

We show that a cellular automaton on a one-dimensional two-sided mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from previous joint work of the author and Törmä that von Neumann regularity is a decidable condition, and we decide it for all elementary CA, obtaining the optimal radii for weak generalized inverses. Two sufficient conditions for non-regularity are having a proper sofic image or having a point in the image with no preimage of the same period. We show that the non-regular ECA 9 and 28 cannot be proven non-regular using these methods. We also show that a random cellular automaton is non-regular with high probability.

An element a in a ring R is called almost clean if a can be written as a = r + e, where r is a regular (non zero-divisor) element and e is an idempotent. It is well known that C(X) is clean if and only if it is almost clean. In this paper a topological characterization of almost clean elements of C(X) is given and using this by a direct and short proof, it is shown that C(X) is clean if and only if it is almost clean. The coincidence of the various kind of cleanness of C(X) is studied. Whenever X is locally compact, we will show that CK(X) is almost clean if and only if CK (X) is clean if and only if C ∞(X) is clean if and only if C ∞(X) is almost clean. It turns out that a prime ideal of C(X) is clean if and only if it is almost clean. We also show that (which in the special case is equal to the super socle of C(X)) is clean and if X is a Lindelof weak P -space, then MβX \\I(X) is almost clean. We prove that X is a P -space if and only if C(X) is a von Neumann u-regular ring (we say that R is a von Neumann u-regular ring if a is a von Neumann regular element implies that 1 + a is a von Neumann regular element, for any a ∈ R). We observe that a ring R is a von Neumann regular ring if and only if it is clean and von Neumann u-regular. Finally, it is shown that if X is a connected space, then X is an almost P -space if and only if every almost clean element of C(X) is clean.

Motivated by some recent work on von Neumann regular elements in semiprime rings, we study how strongly regular elements of semiprime rings are related in terms of their sets of strong inner inverses and strong reflexive inverses.

In a semiprime ring, von Neumann regular elements are determined by their inner inverses. In particular, for elements [Formula: see text] of a von Neumann regular ring [Formula: see text], [Formula: see text] if and only if [Formula: see text], where [Formula: see text] denotes the set of inner inverses of [Formula: see text]. We also prove that, in a semiprime ring, the same is true for reflexive inverses.

Let [Formula: see text] be an associative unital ring with an endomorphism [Formula: see text] and [Formula: see text]-derivation [Formula: see text]. Some types of ring elements such as the units and the idempotents play distinguished roles in noncommutative ring theory, and will play a central role in this work. In fact, we are interested to study the unit elements, the idempotent elements, the von Neumann regular elements, the [Formula: see text]-regular elements and also the von Neumann local elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible. As an application, we completely characterize the clean elements of the Ore extension ring [Formula: see text], when the base ring [Formula: see text] is a right duo ring which is [Formula: see text]-compatible.

Let R be an associative ring with identity and α be an endomorphism of R. In this article, we are interested to study the some of relations between a ring R and that of D. A. Jordan’s construction of the ring A(R,α) as well as the skew Laurent polynomial ring . The main propose of this article is to characterize the unit elements, the idempotent elements, von Neumann regular elements, π-regular elements, von Neumann local elements and also the clean elements of the skew Laurent polynomial ring as well as Jordan’s construction of the ring A(R,α). Applying these characterizations, one can easily get some nice radical-theoretic properties of the mentioned classes of rings.

Many results on going-down domains and divided domains are generalized to the context of rings with von Neumann regular total quotient rings. A (commutative unital) ring R is called regular divided if each P ∈ Spec(R)∖(Max(R) ∩ Min(R)) is comparable with each principal regular ideal of R. Among rings having von Neumann regular total quotient rings, the regular divided rings are the pullbacks K× K/P D where K is von Neumann regular, P ∈ Spec(K) and D is a divided domain. Any regular divided ring (for instance, regular comparable ring) with a von Neumann regular total quotient ring is a weak Baer going-down ring. If R is a weak Baer going-down ring and T is an extension ring with a von Neumann regular total quotient ring such that no regular element of R becomes a zero-divisor in T, then R ⊆ T satisfies going-down. If R is a weak Baer ring and P ∈ Spec(R), then R + PR (P) is a going-down ring if and only if R/P and R P are going-down rings. The weak Baer going-down rings R such that Spec(R)∖Min(R) has a unique maximal element are characterized in terms of the existence of suitable regular divided overrings.

In this article, we provide an alternative approach to the definition of a weak Hopf algebra (WHA). For an associative unital algebra A with a coassociative comultiplication Δ ∈Alg u (A, A ⊗ A), the set of homomorphisms from A to A ⊗ A, which do not preserve the units. If the linear maps Ξ1, Ξ2 ∈ End(A ⊗ A), defined by Ξ1(a ⊗ b) = Δ(a)(1 ⊗ b), Ξ2(a ⊗ b) = (a ⊗ 1)Δ(b), are von Neumann regular elements in the ring End(A ⊗ A) of endomorphisms of A ⊗ A satisfying some appropriate assumptions, we call the A a Hopf-type algebra. We show the existence of a target, a source, a counit, and an antipode of A as in the usual WHA.

Let A and B be reduced archimedean f-rings, A with identity e; let \(A\,\mathop \to \limits^\gamma\,B\) be an l-group homomorphism, and set w = γ (e). We show (with some vagaries of phrasing here) (1) γ = w·ρ for a canonical l-ring homomorphism \(A\,\mathop \to \limits^\rho\,B (w)\), where B (w) is an extension of B in which w is a von Neumann regular element, and (2) for XA,XB canonical representation spaces for A, B, γ is realized via composition with a unique partially defined continuous function from XB to XA.

von Neumann regular elements

Semiabelian rings, defined by the property that each of their idempotents is either left semicentral or right semicentral, are one among several natural generalizations of abelian rings. In this semi-expository paper, we review a number of interesting properties of semiabelian (and other closely allied) rings that are so far well known only for abelian rings. For instance, semiabelian rings [Formula: see text] are always “J-abelian” in the sense that each idempotent of [Formula: see text] maps onto a central idempotent in [Formula: see text]/rad([Formula: see text]). On the other hand, J-abelian rings turn out to be precisely the “strongly perspective rings” as well as the “strongly IC rings”, and the von Neumann regular elements in such rings are automatically strongly regular and are closed under taking [Formula: see text]-th powers. In addition, all J-abelian exchange rings have idempotent stable range one, and are in particular clean (although not necessarily strongly clean) rings.

Let R be an associative ring with unity. An element a ∈ R is said to be r-clean if a = e + r, where e is an idempotent and r is a von Neumann regular element in R. If every element of R is r-clean, then R is called an r-clean ring. In this paper, we investigate the conditions under which the group ring RG is r-clean. We show that if R is a ring and G is a locally finite p-group with p ∈ J(R), then the group ring RG is r-clean if and only if R is r-clean.

We explore aJB*-triple analogue of the notion of quasi invertible elements, originally studied by Brown and Pedersen in the setting ofC*-algebras. This class of BP-quasi invertible elements properly includes all invertible elements and all extreme points of the unit ball and is properly included in von Neumann regular elements in aJB*-triple; this indicates their structural richness. We initiate a study of the unit ball of aJB*-triple investigating some structural properties of the BP-quasi invertible elements; here and in sequent papers, we show that various results on unitary convex decompositions and regular approximations can be extended to the setting of BP-quasi invertible elements. SomeC*-algebra andJB*-algebra results, due to Kadison and Pedersen, Rørdam, Brown, Wright and Youngson, and Siddiqui, including the Russo-Dye theorem, are extended toJB*-triples.