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Suppose that R is an associative unital ring and that $E=(E^0,E^1,r,s)$ is a directed graph. Using results from graded ring theory, we show that the associated Leavitt path algebra $L_R(E)$ is simple if and only if R is simple, $E^0$ has no nontrivial hereditary and saturated subset, and every cycle in E has an exit. We also give a complete description of the centre of a simple Leavitt path algebra. [ABSTRACT FROM AUTHOR]

This paper surveys the theory of functional identities and its applications. No prior knowledge of the theory is required to follow the paper. [ABSTRACT FROM AUTHOR]

For a subset T of a ring R we denote by ℐ (T) the ideal of R generated by T. Given a higher commutator L of R , if R = ℐ (L) then R = L + L 2 ? The question is motivated by the result that a ring R is equal to its subring generated by [ R , R ] if R is either a noncommutative simple ring (by Herstein) or a unital ring with 1 ∈ [ R , R ] (by Eroǧlu). In this note, we study the question for the rings R satisfying the property that every proper ideal of R is contained in a maximal ideal (in particular, if R is finitely generated as an ideal). [ABSTRACT FROM AUTHOR]

A nonzero sum of a unit and a nilpotent element in a ring is called a fine element. This is a study of rings in which every nonzero nilpotent is fine, which we call NF rings. [ABSTRACT FROM AUTHOR]

It is proved that 2-torsion-free simple right-symmetric superrings having a nontrivial idempotent and satisfying a superidentity (x, y, z) + (−1)z(x+y)(z, x, y) + (−1)x(y+z)(y, z, x) = 0 are associative. As a consequence, every simple finitedimensional (1, 1)-superalgebra with semisimple even part over an algebraically closed field of characteristic 0 is associative. [ABSTRACT FROM AUTHOR]

Let Q be a (not necessarily unital) simple ring or algebra. A nonempty subset S of Q is said to have zero product if S 2 = 0. We classify all maximal zero product subsets of Q by proving that the map R ↦ R ∩ LeftAnn (R) is a bijection from the set of all proper nonzero annihilator right ideals of Q onto the set of all maximal zero product subsets of Q. We also describe the relationship between the maximal zero product subsets of Q and the maximal inner ideals of its associated Lie algebra. [ABSTRACT FROM AUTHOR]

A ring is quasi-Baer (respectively, Baer) in case the right annihilator of every ideal (respectively, subset) is generated by an idempotent, as a right ideal. In this article, we investigate on the relationship between the quasi-Baerness, Baerness, semiprimitivity, simplicity and NI properties of a ring R, and its skew Hurwitz series ring (H R , α) , where R is a ring equipped with an endomorphism α . [ABSTRACT FROM AUTHOR]

Given a non-associative unital ring R , a monoid G and a set π of additive maps R → R , we introduce the Ore monoid ring R [ π ; G ] , and, in a special case, the differential monoid ring. We show that these structures generalize, in a natural way, not only the classical Ore extensions and differential polynomial rings, but also the constructions, introduced by Cojuhari, defined by so-called D -structures π. Moreover, for commutative monoids, we give necessary and sufficient conditions for differential monoid rings to be simple. We use this in a special case to obtain new and shorter proofs of classical simplicity results for differential polynomial rings in several variables previously obtained by Voskoglou and Malm by other means. We also give examples of new Ore-like structures defined by finite commutative monoids. [ABSTRACT FROM AUTHOR]

In this paper, we characterize Lie ideals, which are either finitely generated ℤ -modules or maximal, in (centrally closed) prime rings. As consequences, we extend the results proved in [1] for finite dimensional central division algebras of characteristic not 2 to simple rings of arbitrary characteristic. [ABSTRACT FROM AUTHOR]

The Gelfand-Kirillov dimension has gained importance since its introduction as a tool in the study of non-commutative infinite dimensional algebras and their modules. In this paper we show a dichotomy for the Gelfand-Kirillov dimension of simple modules over certain simple rings of differential operators. We thus answer a question of J. C. McConnell in Representations of solvable Lie algebras V. On the Gelfand-Kirillov dimension of simple modules. McConnell (J. Algebra 76(2), 489-493, 1982) concerning this dimension for a class of algebras that arise as simple homomorphic images of solvable lie algebras. We also determine the Gelfand-Kirillov dimension of an induced module. [ABSTRACT FROM AUTHOR]

Weighted Leavitt path algebras (wLpas) are a generalisation of Leavitt path algebras (with graphs of weight 1) and cover the algebras L ( n, n + k) constructed by Leavitt. Using Bergman's diamond lemma, we give normal forms for elements of a weighted Leavitt path algebra. This allows us to produce a basis for a wLpa. Using the normal form we classify the wLpas which are domains, simple and graded simple rings. For a large class of weighted Leavitt path algebras we establish a local valuation and as a consequence we prove that these algebras are prime, semiprimitive and nonsingular but contrary to Leavitt path algebras, they are not graded von Neumann regular. [ABSTRACT FROM AUTHOR]

The Weyl algebra over a field k of characteristic 0 is a simple ring of Gelfand-Kirillov dimension 2, which has a grading by the group of integers. We classify all Z-graded simple rings of GK-dimension 2 and show that they are graded Morita equivalent to generalized Weyl algebras as defined by Bavula. More generally, we study Z-graded simple rings A of any dimension which have a graded quotient ring of the form K[t, t-1; σ] for a field K. Under some further hypotheses, we classify all such A in terms of a new construction of simple rings which we introduce in this paper. In the important special case that GKdimA = tr. deg(K/k) + 1, we show that K and σ must be of a very special form. The new simple rings we define should warrant further study from the perspective of noncommutative geometry. [ABSTRACT FROM AUTHOR]

It is conjectured that every ring with a finite maximal subring is finite. We prove this conjecture for PI-rings.

It is well known that being an ideal (left ideal, right ideal) of a ring is not a transitive relation. Nevertheless in some cases the transitive property does hold. Systematic studies of this subject were started in [11,13]. In this paper we continue these studies. [ABSTRACT FROM AUTHOR]

Let R be a ring. If we replace the original associative product of R with their canonic Lie product, or [a, b] = ab - ba for every a, b in R, then R would be a Lie ring. With this new product the additive commutator subgroup of R or [R, R] is a Lie subring of R. Herstein has shown that in a simple ring R with characteristic unequal to 2, any Lie ideal of R either is contained in Z(R), the center of R or contains [R, R]. He also showed that in this situation the Lie ring [R, R]/Z[R, R] is simple. We give an alternative matrix proof of these results for the special case of simple artinian rings and show that in this case the characteristic condition can be more restricted. [ABSTRACT FROM AUTHOR]

Abstract: We determine simplicity criteria in characteristics 0 and p for a ubiquitous class of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction to simplicity is the possible existence of a canonical normal element z. In the case where this element exists we give simplicity criteria for the rings obtained by inverting z and the rings obtained by factoring out the ideal generated by z. The results are illustrated by numerous examples including higher quantized Weyl algebras and generalizations of some low-dimensional symplectic reflection algebras. [Copyright &y& Elsevier]

Abstract: In this paper we prove three theorems about twisted generalized Weyl algebras (TGWAs). First, we show that each non-zero ideal of a TGWA has non-zero intersection with the centralizer of the distinguished subalgebra R. This is analogous to earlier results known to hold for crystalline graded rings. Second, we give necessary and sufficient conditions for the centralizer of R to be commutative (hence maximal commutative), generalizing a result by V. Mazorchuk and L. Turowska. And third, we generalize results by D.A. Jordan and V. Bavula on generalized Weyl algebras by giving necessary and sufficient conditions for certain TGWAs to be simple, in the case when R is commutative. We illustrate our theorems by considering some special classes of TGWAs and providing concrete examples. We also discuss how simplicity of a TGWA is related to the maximal commutativity of R and the (non-)existence of non-trivial -invariant ideals of R. [Copyright &y& Elsevier]

Abstract: We introduce a new family of twisted generalized Weyl algebras, called multiparameter twisted Weyl algebras, for which we parametrize all simple quotients of a certain kind. Both Jordanʼs simple localization of the multiparameter quantized Weyl algebra and Hayashiʼs q-analog of the Weyl algebra are special cases of this construction. We classify all simple weight modules over any multiparameter twisted Weyl algebra. Extending results by Benkart and Ondrus, we also describe all Whittaker pairs up to isomorphism over a class of twisted generalized Weyl algebras which includes the multiparameter twisted Weyl algebras. [Copyright &y& Elsevier]

We continue our study of the lattice of matric-extensible radicals of associative rings. We find some atoms generated by simple rings of the lattices of all matric-extensible radicals, matric-extensible supernilpotent radicals and matric-extensible special radicals. We consider *-rings, which were previously defined by the second author, and consider when they generate atoms of these lattices. [ABSTRACT FROM AUTHOR]

Abstract: The main aim of this paper is the adaptation of the results of the papers [Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2) (2001) 677–698; Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Finite-dimensional graded simple algebras, preprint, and M.V. Zaĭtsev, S.K. Segal, Finite gradings of simple Artinian rings, Vestnik Moskov. Univ. Ser. I Mat. Mekh., vol. 3, 2001, pp. 21–24, 77 (in Russian); translation in Moscow Univ. Math. Bull. 56 (3) (2001) 21–24] about the structure of graded simple rings from the gradings by groups to the gradings by cancellative semigroups. [Copyright &y& Elsevier]

Abstract: Let D be a division ring with center F and n ⩾1 a natural number. For S ⊆ M n (D) the commuting graph of S, denoted by Γ(S), is the graph with vertex set S⧹Z(S) such that distinct vertices a and b are adjacent if and only if ab = ba. In this paper we prove that if n >2 and are the sets of all non-invertible, nilpotent, idempotent matrices, and involutions, respectively, then for any division ring D, , , , and are connected graphs. We show that if n >2 and is the set of all upper triangular matrices, then for every algebraic division ring D, is a connected graph. Also it is shown that if is the set of all reducible matrices and M n (D) is algebraic over F, then for n >2, is a connected graph. Finally, we prove that for n ⩾2, is a connected graph, where is the ring of real quaternions. [Copyright &y& Elsevier]

Abstract: An infinite-dimensional -graded k-algebra A is called projectively simple if for every nonzero two-sided ideal . We show that if a projectively simple ring A is strongly noetherian, is generated in degree 1, and has a point module, then A is equal in large degree to a twisted homogeneous coordinate ring . Here X is a smooth projective variety, is an automorphism of X with no proper -invariant subvariety (we call such automorphisms wild), and is a -ample line bundle. We conjecture that if X admits a wild automorphism then every irreducible component of X is an abelian variety. We prove several results in support of this conjecture; in particular, we show that the conjecture is true if . In the case where X is an abelian variety, we describe all wild automorphisms of X . Finally, we show that if A is projectively simple and admits a balanced dualizing complex, then is Cohen–Macaulay and Gorenstein. [Copyright &y& Elsevier]