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Peyman Nasehpour

Publications and source records attributed to Peyman Nasehpour.

At least 19 recordsLinked to original sources
arXiv2026

Algebraic properties of overflow semirings

We introduce the overflow semiring $S = A \uplus_{\operatorname{ord}} L$, extending a positive information algebra $A$ by a join-semilattice $L$, where elements of $L$ dominate $A$ and arithmetic in $L$ reduces to the join. This models overflow in computational systems and generalizes the transition from finite to infinite cardinal arithmetic. We characterize the idempotent elements of $S$ and $S[X]$, fully classify idempotent power series over cardinal numbers, describe the structure of prime and maximal ideals, compute the Krull dimension of $S$ ($\dim S = \dim A + |L|$ for well-ordered finite $L$), and establish Noetherian and Artinian criteria.

math.AC↗
arXiv2026

On the distributivity of the subtractive closure operator over ideal operations in hemirings

In this paper, we investigate the subtractive closure operator in commutative hemiring theory. While this operator satisfies standard closure properties, it does not distribute over ideal addition, intersection, or multiplication in general. We introduce Kuratowski hemirings (where closure preserves joins) and nucleus hemirings (where closure preserves meets), providing explicit counterexamples and positive results for each. Key findings include: the standard semiring $\mathbb{N}_0$ and the max-plus algebra are nucleus hemirings with multiplicative distributivity; the interval semiring $T = [0,1]$ is Kuratowski and multiplicatively distributive but not nucleus; and polynomial hemirings $S_0[X,Y,Z]$ over zerosumfree hemirings form a broad class of non-nucleus counterexamples. We also correct a recent claim regarding distributivity over arbitrary intersections.

math.AC↗
arXiv2026

In Memoriam: Professor Dr. Winfried Bruns (1946--2026)

Professor Winfried Bruns was a prominent mathematician with vast knowledge in various fields of mathematics. He specialized in combinatorial commutative algebra, algebraic geometry, computer algebra, algebraic coding theory, and cryptography. In his classes, he incorporated the history of mathematics, as well as topics related to music, making his lectures engaging and enriching from an educational perspective. Additionally, he was a skilled musician, specializing in the viola. The main purpose of this note is to explore his life as a mathematician, computer scientist, and musician.

math.HO↗
arXiv2025

Hemiring-valued pseudonormed rings

In the second section, we introduce hemiring-valued pseudonormed rings and generalize Albert's result which states that every finite-dimensional algebra can be normed. Next, we introduce shrinkable hemirings and prove that dense division semirings are shrinkable. In the third section, we show the Cauchy Condensation Test holds for Cauchy complete fields. In the fourth section, we use Bernoulli's inequality to prove a version of ratio test for ring-valued normed groups.

math.AC↗
arXiv2025

Magma-valued metric spaces

In the second section, we introduce dense unital magmas and show that a near-ring is dense if and only if it has a positive element smaller that unity. In the third section, we discuss magma-valued metric spaces. The density property of the ordered unital magmas and monoids helps us to generalize a couple of classical results related to the convergence and Cauchy property of sequences. The last section is devoted to magma-valued normed groups and some subgroups of the additive group of Cauchy sequences.

math.AC↗
arXiv2025

Covering conditions for ideals in semirings

In this paper, we prove prime avoidance for ringoids. We also generalize McCoy's and Davis' prime avoidance theorems in the context of semiring theory. Next, we proceed to define and characterize compactly packed semirings and show that a commutative semiring is compactly packed if and only if each prime ideal is the radical of a principal ideal. Finally, we calculate the set of zero-divisors of some monoid semimodules over compactly packed semirings in terms of their prime ideals.

math.AC↗
arXiv2025

Algebraic properties of Indigenous semirings

In this paper, we introduce Indigenous semirings and show that they are examples of information algebras. We also attribute a graph to them and discuss their diameters, girths, and clique numbers. On the other hand, we prove that the Zariski topology of any Indigenous semiring is the Sierpiński space. Next, we investigate their algebraic properties (including ideal theory). In the last section, we characterize units and idempotent elements of formal power series over Indigenous semirings.

math.AC↗
arXiv2024

Classical semirings

In this paper, we investigate semirings whose elements are either units or zero-divisors (nilpotents) with many examples. While comparing these semirings with their counterparts in ring theory, we observe that their behavior is different in many cases.

math.AC↗
arXiv2023

Distinguished elements in semiring extensions

In this paper, we investigate zero-divisor, nilpotent, idempotent, unit, small, and irreducible elements in semiring extensions such as amount, content, and monoid semialgebras. We also introduce new concepts such as the prime avoidance property in semirings, entire-like semirings, semialgebras with Property (A), and also, Armendariz and McCoy semialgebras and we prove some results related to these concepts. For example, we prove that if $B$ is an $S$-semialgebra, then under some conditions, the set of zero-divisors $Z(B)$ of $B$ is the union of the extended maximal primes of $Z(S)$. Finally, we prove a generalization of Eisenstein's irreducibility criterion.

math.AC↗
arXiv2020

Amount algebras

In this paper, as a generalization to content algebras, we introduce amount algebras. Similar to the Anderson-Badawi $ω_{R[X]}(I[X])=ω_R(I)$ conjecture, we prove that under some conditions, the formula $ω_B(I^ε)=ω_R(I)$ holds for some amount $R$-algebras $B$ and some ideals $I$ of $R$, where $ω_R(I)$ is the smallest positive integer $n$ that the ideal $I$ of $R$ is $n$-absorbing. A corollary to the mentioned formula is that if, for example, $R$ is a Prüfer domain or a torsion-free valuation ring and $I$ is a radical ideal of $R$, then $ω_{R[][X]]}(I[[X]])=ω_R(I)$.

math.AC↗
arXiv2020

A generalization of zero-divisor graphs

In this paper, we introduce a family of graphs which is a generalization of zero-divisor graphs and compute an upper-bound for the diameter of such graphs. We also investigate their cycles and cores.

math.RA↗
arXiv2020

On Hardy's Apology Numbers

Twelve well known `Recreational' numbers are generalized and classified in three generalized types Hardy, Dudeney, and Wells. A novel proof method to limit the search for the numbers is exemplified for each of the types. Combinatorial operators are defined to ease programming the search.

math.NT↗
arXiv2019

A computational criterion for the irrationality of some real numbers

In this paper, we compute the asymptotic average of the decimals of some real numbers. With the help of this computation, we prove that if a real number cannot be represented as a finite decimal and the asymptotic average of its decimals is zero, then it is irrational. We also show that the asymptotic average of the decimals of simply normal numbers is 9/2.

math.AC↗
arXiv2019

Dedekind semidomains

We define Dedekind semidomains as semirings in which each nonzero fractional ideal is invertible. Then we find some equivalent condition for semirings to being Dedekind. For example, we prove that a Noetherian semidomain is Dedekind if and only if it is multiplication. Then we show that a subtractive Noetherian semidomain is Dedekind if and only if it is a $π$-semiring and each of it nonzero prime ideal is invertible. We also show that the maximum number of the generators of each ideal of a subtractive Dedekind semidomain is 2.

math.RA↗
arXiv2019

On 2-absorbing ideals of commutative semirings

In this paper, we investigate 2-absorbing ideals of commutative semirings and prove that if $\mathfrak{a}$ is a nonzero proper ideal of a subtractive valuation semiring $S$ then $\mathfrak{a}$ is a 2-absorbing ideal of $S$ if and only if $\mathfrak{a}=\mathfrak{p}$ or $\mathfrak{a}=\mathfrak{p}^2$ where $\mathfrak{p}=\sqrt\mathfrak{a}$ is a prime ideal of $S$. We also show that each 2-absorbing ideal of a subtractive semiring $S$ is prime if and only if the prime ideals of $S$ are comparable and if $\mathfrak{p}$ is a minimal prime over a 2-absorbing ideal $\mathfrak{a}$, then $\mathfrak{am} = \mathfrak{p}$, where $\mathfrak{m}$ is the unique maximal ideal of $S$.

math.RA↗
arXiv2019

Eversible and reversible semigroups and semirings

The main purpose of this paper is to investigate the zero-divisors of semigroups with zero and semirings and in particular, to discuss eversible and reversible semigroups and semirings. We also introduce a new ring-like algebraic structure called prenearsemiring and generalize Cohn's theorem for reversible rings in this context. Finally, we discuss when expectation semirings are reversible or eversible.

math.RA↗
arXiv2018

Algebraic properties of expectation semirings

In this paper, we investigate the algebraic properties of the expectation semirings which are semiring version of the concept of trivial extension in ring theory. We discuss ideals, primes, maximals and primary ideals of these semirings. We also discuss the distinguished elements such as the units, idempotents, and zero-divisors of the expectations semirings. Similar to their counterparts in ring theory, we introduce présimplifiable, domainlike, clean, almost clean, and weakly clean semiring and see when an expectation semiring is one of these semirings.

math.AC↗