This text, drawn from the author's lectures at the University of California at Berkeley, is intended as a textbook for a one-term course in basic ring theory. The material covered includes the Wedderburn-Artin theory of semi-simple rings, Jacobson's theory of the radical representation theory of groups and algebras, prime and semi-prime rings, primitive and semi-primitive rings, division rings, ordered rings, local and semi-local rings, and perfect and semi-perfect rings. By aiming the level of writing at the novice rather than at the expert, and by stressing the role of examples and motivation, the author has produced a text which is suitable not only for use in a graduate course, but also for self-study by other interested graduate students. Numerous exercises are also included. This graduate textbook on rings, fields and algebras is intended for graduate students in mathematics.

A first course in noncommutative rings, by T. Y. Lam. Pp. 385. £37 (pb), £62.50 (hb). 2001. ISBN 0 387 95325 6 (pb), 0 387 95183 0 (hb) (Springer-Verlag).

The Algebraic Theory of Semigroups, Vol. I. By A. H. Clifford and G. B. Preston. Pp. xv + 224. $10.60. 1961. (American Mathematical Society, Providence.)

The aim of this paper is to introduce some new operators for aggregating single-valued neutrosophic (SVN) information and to apply them to solve the multi-criteria decision-making (MCDM) problems. Single-valued neutrosophic set, as an extension and generalization of an intuitionistic fuzzy set, is a powerful tool to describe the fuzziness and uncertainty, and Muirhead mean (MM) is a well-known aggregation operator which can consider interrelationships among any number of arguments assigned by a variable vector. In order to make full use of the advantages of both, we introduce two new prioritized MM aggregation operators, such as the SVN prioritized MM (SVNPMM) and SVN prioritized dual MM (SVNPDMM) under SVN set environment. In addition, some properties of these new aggregation operators are investigated and some special cases are discussed. Furthermore, we propose a new method based on these operators for solving the MCDM problems. Finally, an illustrative example is presented to testify the efficiency and superiority of the proposed method by comparing it with the existing method.

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Special Issue: Algebraic Structures of Neutrosophic Triplets, Neutrosophic Duplets, or Neutrosophic Multisets, Vol. II

Let R be a prime algebra over a commutative ring K with unity and let f(X1, . . . ,Xn) be a multilinear polynomial over K. Suppose that d is a nonzero derivation on R such that for all r1, . . . , rn in some nonzero ideal I of R, [ d ( f(r1, . . . , rn) ) , f(r1, . . . , rn) ] k = 0 with k fixed. Then f(X1, . . . , Xn) is central–valued on R except when char R = 2 and R satisfies the standard identity s4 in 4 variables. Throughout this note K will denote a commutative ring with unity and R will denote a prime K–algebra with center Z. By d we always mean a nonzero derivation onR. For x, y ∈ R, set [x, y]0 = x, [x, y]1 = [x, y] = xy−yx and, for k > 1, [x, y]k = [ [x, y]k−1, y ] . A well–known result proved by Posner [10] states that R must be commutative if [ d(x), x ] ∈ Z for all x ∈ R. In [7], the authors generalized Posner’s theorem by showing that a Lie ideal L of R must be contained in Z if char R 6= 2 and [ d(x), x ] ∈ Z for all x ∈ L. As to the case when char R = 2, Lanski [5] obtained the same conclusion except when R satisfies the standard identity s4 in 4 variables. On the other hand, Vukman [11] showed that R is commutative if char R 6= 2 and [ d(x), x ] 2 = 0 for all x ∈ R, or if char R 6= 2, 3 and [ d(x), x ] 2 ∈ Z for all x ∈ R. In a recent paper [6], a full generalization of these results was proved by Lanski. He showed that a Lie ideal L of R is in Z if for some fixed k > 0, [ d(x), x ] k = 0 for all x ∈ L, unless char R = 2 and R satisfies s4. Note that a noncentral Lie ideal of R contains all the commutators [x, y] for x, y in some nonzero ideal of R except when char R = 2 and R satisfies s4. It is natural to consider the situation when [ d(x), x ] k = 0 for all commutators x = [x1, x2], or more generally, when [ d(x), x ] k = 0 for all x = f(x1, . . . , xn) where f(X1, . . . , Xn) is a multilinear polynomial over K. In the present paper, we shall extend Lanski’s theorem by imposing the condition [ d ( f(x1, . . . , xn) ) , f(x1, . . . , xn) ] k = 0 on some nonzero ideal of R. First we dispose of the simplest case when R is the matrix ring Mm(F ) over a field F and d is an inner derivation on R. Received by the editors November 4, 1994 and, in revised form, March 1, 1995. 1991 Mathematics Subject Classification. Primary 16W25; Secondary 16N60, 16R50, 16U80.

Derivations with engel conditions on multilinear polynomials

We introduce the notions of quasi-ideal and bi-ideal in ternary semirings and study some properties of these two ideals. We also characterize regular ternary semiring in terms of these two subsystems of ternary semirings.

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On quasi-ideals and bi-ideals in ternary semirings

On Regular Ternary Semirings

On prime ideals and prime radical of ternary semiring

This paper is a sequel of our previous paper (1). In this paper, we introduce the notions of regular ideal and partial ideal (p-ideal) in a ternary semiring and using these two notions we characterize regular ternary semiring.

Sukhendu Kar

Papers

On Regular Ternary Semirings

On prime ideals and prime radical of ternary semiring

On quasi-ideals and bi-ideals in ternary semirings

A Note on Regular Ternary Semirings

On semiprime ideals and irreducible ideals of ternary semiring