Industrial Engineering & Management Systems

ABSTRACT. Let M be an R-module and c the function from M to the ideals of R defined by c(x) = ∩{I : I is an ideal of R and x ∈ IM}. M is said to be a content R-module if x ∈ c(x)M, for all x ∈ M. B is called a content R-algebra, if it is a faithfully flat and content R-module and ...

Throughout this paper all rings are commutative with unit and all modules are assumed to be unitary. In this paper, we discuss zero-divisors of content algebras. To this end, one needs to know about content modules and algebras introduced in [19]. Our main goal is to show ...

We find strong relationships between the zero-divisor graphs of apparently disparate kinds of nilpotent-free semigroups by introducing the notion of an \emph{Armendariz map} between such semigroups, which preserves many graph-theoretic invariants. We use it to give relationships between the zero-divisor graph of a ring, a polynomial ring, and the annihilating-ideal graph. Then we give relationships between the zero-divisor graphs of certain topological spaces (so-called pearled spaces), prime spectra, maximal spectra, tensor-product semigroups, and the semigroup of ideals under addition, obtaining surprisingly strong structure theorems relating ring-theoretic and topological properties to graph-theoretic invariants of the corresponding graphs.

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.

Let R be a commutative ring with identity. An ideal a of R is called a cancellation ideal if whenever ab= ac for ideals b and c of R, then b= c. A good introduction to cancellation ideals may be found in Gilmer [[3]; section 6]. An R–module M is called cancellation module, if whenever aM= bM for ideals a and b of R, then a= b, cf.[6]. The ideal a of R is called an M-cancellation ideal if whenever aP= aQ for submodules P and Q of M, then P= Q. In the first section we characterize M-cancellation ideals.

The main task 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 iff 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 iff 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 iff its unique maximal ideal is subtractive.

In this paper, we define finitely additive, probability and modular functions over semiring-like structures. We investigate finitely additive functions with the help of complemented elements of a semiring. Then with the help of those observations, we also generalize some classical results in probability theory such as Boole's Inequality, the Law of Total Probability, Bayes' Theorem, the Equality of Parallel Systems, and Poincar\'{e}'s Inclusion-Exclusion Theorem. While we prove that modular functions over a couple of celebrated semirings are almost constant, we show it is possible to define many different modular functions over some semirings such as bottleneck algebras and the semiring (Id(D),+,⋅), where D is a Dedekind domain. Finally, we prove that under suitable conditions a function f is finitely additive iff it is modular and f(0)=0.

In the first section, we introduce the notions of fractional and invertible ideals of semirings and characterize invertible ideals of a semidomain. In section two, we define Prüfer semirings and characterize them in terms of valuation semirings. In this section, we also characterize Prüfer semirings in terms of some identities over its ideals such as (I + J)(I ∩ J) = IJ for all ideals I, J of S. In the third section, we give a semiring version for the Gilmer-Tsang Theorem, which states that for a suitable family of semirings, the concepts of Prüfer and Gaussian semirings are equivalent. At last, we end this paper by giving a plenty of examples for proper Gaussian and Prüfer semirings.

The late Professor Ali Reza Zokayi (1944--2018) was a group theorist and a student of Prof. Dr. Zvonimir Janko. The contribution of Zokayi was essential for the discovery of new finite simple groups and this was of great importance for the classification of finite simple groups.