Peyman Nasehpour

Bio: Peyman Nasehpour is an academic researcher from Golpayegan University of Engineering. The author has contributed to research in topics: Semiring & Zero divisor. The author has an hindex of 8, co-authored 52 publications receiving 231 citations. Previous affiliations of Peyman Nasehpour include University of Osnabrück & University of Tehran.

On the content of polynomials over semirings and its applications

Abstract: In this paper, we prove that Dedekind–Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a content semialgebra over a semiring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also discuss formal power series semirings and show that under suitable conditions, they are good examples of weak content semialgebras.

Valuation Semirings

Abstract: The main scope of this paper is to introduce valuation semirings in general and discrete valuation semirings in particular In order to do that, first we define valuation maps and investigate them Then we define valuation semirings with the help of valuation maps and prove that a multiplicatively cancellative semiring is a valuation semiring if and only if its ideals are totally ordered by inclusion We also prove that if the unique maximal ideal of a valuation semiring is subtractive, then it is integrally closed We end this paper by introducing discrete valuation semirings and show that a semiring is a discrete valuation semiring if and only if it is a multiplicatively cancellative principal ideal semiring possessing a nonzero unique maximal ideal We also prove that a discrete valuation semiring is a Gaussian semiring if and only if its unique maximal ideal is subtractive

Zero-divisors of content algebras

Abstract: In this article, we prove that in content extentions minimal primes extend to minimal primes and discuss zero-divisors of a content algebra over a ring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also examine the preservation of diameter of zero-divisor graph under content extensions.

On the Content of Polynomials Over Semirings and Its Applications

Abstract: In this paper, we prove that Dedekind-Mertens lemma holds only for those semimodules whose subsemimodules are subtractive. We introduce Gaussian semirings and prove that bounded distributive lattices are Gaussian semirings. Then we introduce weak Gaussian semirings and prove that a semiring is weak Gaussian if and only if each prime ideal of this semiring is subtractive. We also define content semialgebras as a generalization of polynomial semirings and content algebras and show that in content extensions for semirings, minimal primes extend to minimal primes and discuss zero-divisors of a content semialgebra over a semiring who has Property (A) or whose set of zero-divisors is a finite union of prime ideals. We also discuss formal power series semirings and show that under suitable conditions, they are good examples of weak content semialgebras.

Zero-divisors of Semigroup Modules

Abstract: Let M be an R-module and S a semigroup. Our goal is to discuss zero-divisors of the semigroup module M(S). Particularly we show that if M is an R-module and S a commutative, cancellative and torsion-free monoid, then the R(S)-module M(S) has few zero-divisors of size n if and only if the R-module M has few zero-divisors of size n and Property (A).

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).

Abstract: 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.