On zero-dimensionality and fragmented rings

Assignable polynomials to linear systems over von Neumann regular rings

Following O'Meara's result [Journal of Algebra and Its Applications, 13 (2014) 8], it follows that the block matrix A = B 0 0 0 M n+r (R), B M n (R), r 1, over a von Neumann regular separative ring R, is a product of idempotent matrices.Furthermore, this decomposition into idempotents of A also holds when B is an invertible matrix and R is a GE ring (defined by Cohn [New mathematical monographs: 3, Cambridge University Press, 2006]).As a consequence, it follows that if there exists an example of a von Neumann regular ring R over which the matrix A = B 0 0 0 M n+r (R), where B M n (R), r 1, cannot be expressed as a product of idempotents, then R is not separative, thus providing an answer to an open question whether there exists a von Neumann regular ring which is not separative.The paper concludes with a counter example related to an open question whether every totally nonnegative matrix is a product of nonnegative idempotent matrices.

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).

In part I we introduce vertex rings , which bear the same relation to vertex algebras (or VOAs) as commutative, associative rings do to commutative, associative algebras over C \mathbf {C} . We show that vertex rings are characterized by Goddard axioms . These include a generalization of the translation-covariance axiom of VOA theory that involves a canonical Hasse-Schmidt derivation naturally associated to any vertex ring. We give several illustrative applications of these axioms, including the construction of vertex rings associated with the Virasoro algebra. We consider some categories of vertex rings, and the rôle played by the center of a vertex ring. In part II we extend the theory of Pierce bundles associated to a commutative ring to the setting of vertex rings. This amounts to the construction of certain reduced étale bundles of vertex rings functorially associated to a vertex ring. We introduce von Neumann regular vertex rings as a generalization of von Neumann regular commutative rings; we obtain a characterization of this class of vertex rings as those whose Pierce bundles are bundles of simple vertex rings.

Reverse decomposition of unipotents over noncommutative rings I: General linear groups

Let R be a commutative ring with identity. In this paper, we introduce the concept of (m, n)-closed ideals of R and (m, n)-von Neumann regular rings

Let R be a commutative ring with identity. An R-module M is said to be w-projective if <TEX>$Ext\frac{1}{R}$</TEX>(M,N) is GV-torsion for any torsion-free w-module N. In this paper, we define a ring R to be w-semi-hereditary if every finite type ideal of R is w-projective. To characterize w-semi-hereditary rings, we introduce the concept of w-injective modules and study some basic properties of w-injective modules. Using these concepts, we show that R is w-semi-hereditary if and only if the total quotient ring T(R) of R is a von Neumann regular ring and <TEX>$R_m$</TEX> is a valuation domain for any maximal w-ideal m of R. It is also shown that a connected ring R is w-semi-hereditary if and only if R is a Pr<TEX>$\ddot{u}$</TEX>fer v-multiplication domain.

Almost slender rings and compact rings

Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

A characterization of von Neumann rings in terms of linear systems

We show that every flat semimodule having a projective cover coincides with that cover up to isomorphism. Also, we prove that all (finitely generated) semimodules over a semiring [Formula: see text] are flat iff [Formula: see text] is a von Neumann regular ring.

Here are some applications of the theory of the previous chapter. In the first section, it is proven that, for every Baer–Stone half-shell that is two-sided and unital, there is a reduced and factor-transparent sheaf over a Boolean space that represents the half-shell and and has stalks with no divisors of zero. With all that has been done in previous chapters, the proof is relatively short. Just as the results of the previous chapters may be cast into categories, so we restate this result as the equivalence of two categories. In the second section are two more applications. Each von Neumann regular, commutative and unital ring is isomorphic to the ring of all global sections of a sheaf of fields over a Boolean space. And every biregular ring is isomorphic to the ring of all global sections of a sheaf with simple stalks over a Boolean space. These results extend to half-shells.

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ I (R), is the graph whose vertices are {x ∈ R∖I | xy ∈ I for some y ∈ R∖I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ Ann(M)(R).

If x is an element of a commutative ring R then define the x-divisor graph \(\varGamma _x(R)\) to be the graph, whose vertices are the elements of \(d(x)=\{r\in R\) | \(rs=x\) for some \(s\in R\}\) such that two distinct vertices r and s are adjacent if and only if \(rs=x\). In this chapter, the components of \(\varGamma _x(R)\) are completely characterized when R is a von Neumann regular ring. Various other types of “divisor graphs” are considered as well. For example, if x is a nonzero element of an integral domain R with group of units U(R) then the compressed divisor graph \((\varGamma _E)_x^{d^\times }(R)\) associated with x is defined to be the graph, whose vertices are the associate-equivalence classes \(\overline{r}=rU(R)\) of elements \(r\in d(x)^\times =d(x)\setminus (xU(R)\cup U(R))\) such that two distinct vertices \(\overline{r}\) and \(\overline{s}\) are adjacent if and only if \(rs\in d(x)\). Alternatively, by letting M be the positive cone of the group of divisibility of R, every \((\varGamma _E)_x^{d^\times }(R)\) is a member of the class of graphs \(\varGamma _{\le x}(M)\) defined by picking an element x of a partially ordered commutative monoid M with least element equal to its identity 1, and letting the vertices of \(\varGamma _{\le x}(M)\) be the elements of \(\{m\in M\) | \(1<m<x\}\) such that two distinct vertices m and n are adjacent if and only if \(mn\le x\). Other aspects of the chapter include the exploration of graph-theoretic criteria that reveal when two elements of an integral domain are associates, and it is proved that R is a unique factorization domain if and only if \((\varGamma _E)_x^{d^\times }(R)\) is either null or finite with a dominant clique for every \(x\in R\setminus \{0\}\). Throughout, emphasis is placed on similarities with zero-divisor graphs. For example, it is proved that if R is von Neumann regular and G is a component of \(\varGamma _x(R)\) that contains a square root of x then \(G\cong \varGamma _0(\text {ann}_R(x))\) (in particular, if \(x=0\) then we have the tautology \(G\cong \varGamma _0(R)\)), and if x is a square-free element of a unique factorization domain then \((\varGamma _E)_x^{d^\times }(R)\) is isomorphic to a zero-divisor graph of a finite Boolean ring.

A ring R has the (A)-property (resp., strong (A)-property) if every finitely generated ideal of R consisting entirely of zero divisors (resp., every finitely generated ideal of R generated by a finite number of zero-divisors elements of R) has a nonzero annihilator. The class of commutative rings with property (A) is quite large; for example, Noetherian rings, rings whose prime ideals are maximal, the polynomial ring R[x] and rings whose total ring of quotients are von Neumann regular. Let f : A → B be a ring homomorphism and J be an ideal of B. In this paper, we investigate when the (A)-property and strong (A)-property are satisfied by the amalgamation of rings denoted by A ⋈fJ, introduced by D'Anna, Finocchiaro and Fontana in [3]. Our aim is to construct new original classes of (A)-rings that are not strong (A)-rings, (A)-rings that are not Noetherian and (A)-rings whose total ring of quotients are not Von Neumann regular rings.