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The reversible property of rings was introduced by Cohn and has important generalizations in noncommutative ring theory. In this paper, reversibility of rings is investigated in relation with quasinilpotents and idempotents, and our argument is spread out based on this. We call a ring RQnil e-reversible if for any a , b ∈ R , being a ⁢ b = 0 implies b ⁢ a ⁢ e ∈ R qnil for a prescribed idempotent e ∈ R , where R qnil denotes the set of all quasinilpotent elements of R. In the first, we determine the set R qnil for some classes of rings to investigate the structure of Qnil e-reversible rings. In the second, we use R qnil to define Qnil e-reversibility of rings. The notion of Qnil e-reversible ring is a proper generalization of that of e-semicommutative ring, Qnil-semicommutative ring, e-reversible ring and right (left) quasi-duo ring. We obtain some relations between a ring and its quotient rings in terms of Qnil e-reversibility. Applications via some ring extensions and examples illustrating our results are provided. [ABSTRACT FROM AUTHOR]

The main achievement of this paper is a geometric characterisation of certain subvarieties of the Cartan variety 퓔6(핂) over an arbitrary field 핂. The characterised varieties arise as Veronese representations of certain ring projective planes over quadratic subalgebras of the split octonions 핆' over 핂 (among which the sextonions, a 6-dimensional non-associative algebra). We describe how these varieties are linked to the Freudenthal–Tits magic square, and discuss how they would even fit in, when also allowing the sextonions and other "degenerate composition algebras" as the algebras used to construct the square. [ABSTRACT FROM AUTHOR]

By Lemma 2 Graph HT ht where I d i ( I r i SB I h i sb ) I d i ( I h i ) - 2. I Then for any tuple a i HT ht Graph HT ht I where r SB g sb i I PF i SB 2 sb [ I x i SB 1 sb , ..., I x i SB I n i sb ] I and d i ( I r i SB I g i sb ) I d i ( I g i ) - 2. [Extracted from the article]

Let R be an associative ring. A multiplicative semi-derivation d is a map on R satisfying d ⁢ (x ⁢ y) = d ⁢ (x) ⁢ g ⁢ (y) + x ⁢ d ⁢ (y) = d ⁢ (x) ⁢ y + g ⁢ (x) ⁢ d ⁢ (y) and d ⁢ (g ⁢ (x)) = g ⁢ (d ⁢ (x)) {d(xy)=d(x)g(y)+xd(y)=d(x)y+g(x)d(y)\quad\text{and}\quad d(g(x))=g(d(x))} for all x , y ∈ R {x,y\in R} , where g is any map on R. In this paper, we have obtained some conditions on R, which make d additive. Finally, we have also shown that every multiplicative semi-derivation on M n ⁢ (ℂ) {M_{n}(\mathbb{C})} , the algebra of all n × n {n\times n} matrices over the field ℂ {\mathbb{C}} of complex numbers, is an additive derivation. [ABSTRACT FROM AUTHOR]

For four wide classes of topological rings R \mathfrak{R} , we show that all flat left R \mathfrak{R} -contramodules have projective covers if and only if all flat left R \mathfrak{R} -contramodules are projective if and only if all left R \mathfrak{R} -contramodules have projective covers if and only if all descending chains of cyclic discrete right R \mathfrak{R} -modules terminate if and only if all the discrete quotient rings of R \mathfrak{R} are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings. The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes. The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals. [ABSTRACT FROM AUTHOR]

Let ℛ be a prime ring with center Z (ℛ) and * an involution of ℛ. Suppose that ℛ admits generalized derivations F, G and H associated with a nonzero derivation f, g and h of ℛ, respectively. In the present paper, we investigate the commutativity of a prime ring ℛ satisfying any of the following identities: : (i) [F(x), F(x∗)] = 0, (ii) [F(x), F(x∗)] = ±[x, x∗], (iii) F(x) ∘ F(x∗) = 0, (iv) F(x) ∘ F(x∗) = ±(x ∘ x∗), (v) [F(x), x∗] ± [x, G(x∗)] = 0, (vi) F(xx∗) ∈ Z(R), (vii) F(x)G(x∗) ± H(x)x∗ ∈ Z(R), (viii) F([x, x∗]) ± [x, x∗] ∈Z(R), (ix) F(x ∘ x∗) ± x ∘ x∗ ∈ Z(R), (x) [F(x), x∗] ± [x, G(x∗)] ∈ Z(R), (xi) F(x) ∘ x∗ ± x ∘ G(x∗) ∈ Z(R) for all x ∈ ℛ. Finally, the restrictions imposed on the hypotheses have been justified by an example. [ABSTRACT FROM AUTHOR]

We consider the transformations of random variables over a finite associative ring by the addition and multiplication operations. For arbitrary finite rings, we construct families of distribution algebras, which are sets of distributions closed over sums and products of independent random variables. [ABSTRACT FROM AUTHOR]

We extend to arbitrary rings a definition of the octonion special linear group due to Baez. At the infinitesimal level, we get a Lie ring, which we describe over some large classes of rings, including all associative rings and all algebras over a field. As a corollary, we compute all the groups Baez defined. [ABSTRACT FROM AUTHOR]

Centrally essential rings were defined earlier for associative unital rings; in this paper, we define them for rings which are not necessarily associative or unital. In this case, it is proved that centrally essential semiprime rings are commutative. It is proved that all idempotents of a centrally essential alternative ring are central. Several examples of non-commutative centrally essential rings are provided, some properties of centrally essential rings are described. [ABSTRACT FROM AUTHOR]

In the present paper, we introduce the notion of generalized multiplicative derivation in a near-ring N and investigate commutativity of 3-prime near-rings, showing that certain conditions involving generalized multiplicative derivations force N to be a commutative ring. Finally some more results related with the structure of these derivations are also obtained. [ABSTRACT FROM AUTHOR]

In [Acta Math. Hungar. 66 (1995), 337-343], Bell and Daif proved that if R is a prime ring admitting a nonzero derivation such that for all , then R is commutative. The objective of this paper is to examine similar problems when the ring R is equipped with involution. It is shown that if a prime ring R with involution * of a characteristic different from 2 admits a nonzero derivation d such that for all x ∈ R and , then R is commutative. Moreover, some related results have also been discussed. [ABSTRACT FROM AUTHOR]

Let R be a ring (associative, with 1), and let R〈〈 a, b〉〉 denote the power-series R-ring in two non-commuting, R-centralizing variables, a and b. Let A be an R-subring of R〈〈 a〉〉 and B be an R-subring of R〈〈 b〉〉. Let α denote the natural map A ⨿ R B → R〈〈 a, b〉〉. This article describes some situations where α is injective and some where it is not. We prove that if A is a right Ore localization of R[ a] and Bis a right Ore localization of R[ b], then α is injective. For example, the group ring over R of the free group on {1+ a,1+ b} is R[(1+ a)±] ⨿ R R[(1+ b)±], which then embeds in R〈〈 a, b〉〉. We thus recover a celebrated result of R. H. Fox, via a proof simpler than those previously known. We show that α is injective if R is Π- semihereditary, that is, every finitely generated, torsionless, right R-module is projective. (This concept was first studied by M. F. Jones, who showed that it is left-right symmetric. It follows from a result of I. I. Sahaev that if w.gl.dim R ≤ 1 and R embeds in a skew field, then R is Π-semihereditary. Also, it follows from a result of V. C. Cateforis that if R is right semihereditary and right self-injective, then R is Π-semihereditary.) The arguments and results extend easily from two variables to any set of variables. The article concludes with some results contributed by G. M. Bergman that describe situations where α is not injective. He shows that if R is commutative and w.gl.dim R ≥ 2, then there exist examples where the map α': A ⨿ R B → R〈〈 a〉〉 ⨿ R R〈〈 b〉〉 is not injective, and hence neither is α. It follows from a result of K. R. Goodearl that when R is a commutative, countable, non-self-injective, von Neumann regular ring, then the map α'': R〈〈 a〉〉 ⨿ R R〈〈 b〉〉 → R〈〈 a, b〉〉 is not injective. Bergman gives procedures for constructing other examples where α'' is not injective. [ABSTRACT FROM AUTHOR]

We investigate the algebraic K- and L-theory of the group ring RG, where G is a hyperbolic or virtually finitely generated abelian group and R is an associative ring with unit. [ABSTRACT FROM AUTHOR]

We show, for a noetherian ring R, that there are no unexpected colocalizing subcategories of D( R). The only colocalizing subcategories are of the form , where is a localizing subcategory. [ABSTRACT FROM AUTHOR]

We prove that for a noetherian semilocal ring R with exactly k isomorphism classes of simple right modules the monoid V*( R) of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of V*( R/ J( R)), is isomorphic to the monoid of solutions in (ℕℕ0 ∪∪ {∞∞}) k of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if M is a submonoid of (ℕℕ0 ∪∪ {∞∞}) k containing an order unit ( n1, . . . , nk) of which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as V*( R) for a noetherian semilocal ring such that R/ J( R) ≅≅ Mn1( D1) ×× ⋯⋯ ×× Mnk ( Dk) for suitable division rings D1, . . . , Dk. [ABSTRACT FROM AUTHOR]

We consider the near-rings generated by endomorphisms of some extra-special 2-groups. The most essential difference of a near-ring from a usual ring is the absence of the second distributivity. In this paper, we prove that the near-ring E( G) generated by endomorphisms of an extra-special 2-group G of order 22 n+1 has the order which divides 222 n+4 n2 and that the near-ring E( G) of the extra-special 2-group G of type – of order 22 n+1 has the order divided by 222 n+4 n2–2. In this case, for n = 1 and n = 2 the upper bound is attainable: the near-ring E( G) of the group D8 has the order 28, and the near-ring E( G) of an extra-special 2-group D8 * Q8 has the order 232. [ABSTRACT FROM AUTHOR]

In this paper we give necessary and sufficient conditions on a row-finite graph E so that the corresponding (not necessarily unital) Leavitt path K-algebra LK( E) is either artinian or noetherian from both a local and a categorical perspective. These extend the known results in the unital case to a much wider context. Besides the graph theoretic conditions, we provide in both situations isomorphisms between these algebras and appropriate direct sums of matrix rings over K or K[ x, x–1]. [ABSTRACT FROM AUTHOR]

All finitely generated right A-modules are ℵ0-injective if and only if A is a regular, right ℵ0-injective ring. [ABSTRACT FROM AUTHOR]

The zero-divisor graph of an associative ring R is a graph whose vertices are all nonzero (one-sided and two-sided) zero divisors of R, two distinct vertices x, y are connected by an edge if and only if xy = 0 or yx = 0. In this paper, all finite nonnilpotent rings with planar zero-divisor graphs are completely described. In the previous paper by Kuzmina and Maltsev, the finite nilpotent rings with planar zero-divisor graphs were studied. Thus, this paper completes the description of finite rings with planar zero-divisor graphs. [ABSTRACT FROM AUTHOR]

We give a characterization of Σ-cotorsion modules over valuation domains in terms of descending chain conditions on certain chains of definable subgroups. We prove that pure submodules, direct products and modules elementarily equivalent to a Σ-cotorsion module are again Σ-cotorsion. Moreover, we describe the structure of Σ-cotorsion modules. [ABSTRACT FROM AUTHOR]

Let R be an integral domain with quotient field Q. For R-submodules X and Y of Q denote by [ Y : X] the R-module . An ideal I of R is a colon-splitting ideal of R if for all ideals J and K of R. We examine colon-splitting ideals and use these results to characterize a special class of colon-splitting ideals, the ideals of injective dimension 1. We show that every nonzero ideal of a domain is a colon-splitting ideal (has injective dimension 1) if and only if every maximal ideal is a colon-splitting ideal (resp., has injective dimension 1). From this we deduce new characterizations of h-local Prfer domains and almost maximal Prfer domains. [ABSTRACT FROM AUTHOR]

Let A and B be two rings, M be a left B, right A bimodule and be the formal triangular matrix ring. It is known that the category of right T-modules is equivalent to the category Ω of triples ( X, Y) f, where X is a right A-module, Y is a right B-module and is a right A-homomorphism. Using this alternative description of right T-modules, in the first part of this paper, we study the vanishing of the extension functor ‘Ext’ over T. To this end, we first describe explicitly the structure of (right) T-modules of finite projective (respectively, injective) dimension. Using these results, we shall characterize respectively modules in Auslander's G-class, Gorenstein injective modules, cotorsion modules and tilting and cotilting modules over T. As another application we investigate the structure of the flat covers of right T-modules. [ABSTRACT FROM AUTHOR]

We apply tilting theory to study modules of finite projective dimension. We introduce the notion of finite and cofinite type for tilting and cotilting classes of modules, respectively, showing that, for each dimension, there is a bijection between these classes and resolving classes of modules. We then focus on Iwanaga-Gorenstein rings. Using tilting theory, we prove the first finitistic dimension conjecture for these rings. Moreover, we characterize them among noetherian rings by the property that Gorenstein injective modules form a tilting class. Finally, we give an explicit construction of families of (co)tilting modules of (co)finite type for one-dimensional commutative Gorenstein rings. [ABSTRACT FROM AUTHOR]

Let T be a right chain ring with non-zero maximal ideal J . In this paper we study subrings R of T containing J and determine conditions for R to be a right distributive (right Bezout) ring. As a consequence we obtain a structure theorem for semiprime right distributive (right Bezout) rings with non-zero left waists. [ABSTRACT FROM AUTHOR]

Necessary conditions for solvability of systems of linear equations over associative rings are considered. In some cases, these conditions are also sufficient for solvability of systems. This research was supported by the grant 2358.2003.9 of the President of Russian Federation for supporting the leading scientific schools. [ABSTRACT FROM AUTHOR]

This article provides an almost self-contained, purely combinatorial local recognition of the graph on the non-intersecting line-hyperline pairs of the projective space P[subn](F) for n ≥8 and F a division ring with the exception of the case n = 8 and F = F[sub2]. Consequences of that result are a characterization of the hyperbolic root group geometry of SL[subn+1] (F), F a division ring, and a local recognition of certain groups containing a central extension of PSL[subn+1](F), F a field, using centralizers of p-elements. [ABSTRACT FROM AUTHOR]

For the vector space V over the division ring D, let FEnd[subD](V) be the set of all Dtransformations x ∈ End[subD](V) such that x has finite rank, and let FGL[subD] (V) be the set of all g ∈ GL[subD] (V) such that g -- 1 has finite rank. We study subrings A of FEnd[subD](V) and the relationships of such rings to subgroups G of FGL[subD](V) with the additional assumption that D is finite dimensional over its center K. The division ring D is identified with D · id[subv]. With D, A as above, we define D[A] to be the subring of End[subk](V) generated by D and A, and D[G] to be the subring of End[subK](V) generated by D and G. One of our ring-theoretic results is the following: if D[A] is semisimple and M is a right D[A]-module, then M(DA) is a completely reducible D[A]-module. This implies the following group-theoretic result: if G ≤ FGL[subD](V) and G has a local system of subgroups F such that H e F implies that V is a completely reducible D[H]module, then [V, G] is a completely reducible D[G]-module. Several additional known results due to Meierfrankenfeld and Wehrfritz follow naturally as consequences of the ring-theoretic approach adopted here. [ABSTRACT FROM AUTHOR]

Immediate extensions of valued fields are generalized to arbitrary commutative rings. Linearly compact rings and von Neumann regular rings are maximally complete. A necessary and sufficient condition is given for the linear compactness of any maximal immediate extension of a valuation ring. Newton's Approximation Method is used in finding simple roots of polynomials over linearly compact local rings. Kaplansky's method [8], as a non-linear version of Newton's Approximation, is used to find (possibly multiple) roots of polynomials, giving an extension of MacLane's characterization of algebraically closed maximal valued fields to linearly compact valuation rings. [ABSTRACT FROM AUTHOR]