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12 pages, 282 KB

Open AccessArticle

Projective Modules and Polynomials over the Hurwitz Quaternions

by Francis E. A. Johnson

Mathematics 2026, 14(5), 861; https://doi.org/10.3390/math14050861 - 3 Mar 2026

Viewed by 388

A theorem of Sheshadri (Proc. Nat. Acad. Sci. USA 44 (1958) 456–458) shows that, when

Λ

is a commutative principal ideal domain, finitely generated projective modules over the polynomial ring

Λ [t]

are all free. The ring

Γ

of Hurwitz quaternions, [...] Read more.

A theorem of Sheshadri (Proc. Nat. Acad. Sci. USA 44 (1958) 456–458) shows that, when

Λ

is a commutative principal ideal domain, finitely generated projective modules over the polynomial ring

Λ [t]

are all free. The ring

Γ

of Hurwitz quaternions, that is, the unique maximal order in the ring of rational quaternions, is the simplest example of a non-commutative principal ideal domain which is not a division ring. In contrast to the commutative case, we show that

Γ [t]

has infinitely many isomorphically distinct projective modules; these are stably free of rank 1. Full article

18 pages, 304 KB

Open AccessArticle

Additive Biderivations of Incidence Algebras

by Zhipeng Guan and Chi Zhang

Mathematics 2025, 13(19), 3122; https://doi.org/10.3390/math13193122 - 29 Sep 2025

Viewed by 594

Abstract

We characterize all additive biderivations on the incidence algebra

I (P, R)

of a locally finite poset P over a commutative ring with unity

R

. By decomposing P into its connected chains, we prove that any additive biderivation splits [...] Read more.

We characterize all additive biderivations on the incidence algebra

I (P, R)

of a locally finite poset P over a commutative ring with unity

R

. By decomposing P into its connected chains, we prove that any additive biderivation splits uniquely into a sum of inner biderivations and extremal ones determined by chain components. In particular, when every maximal chain of P is infinite, all additive biderivations are inner. Full article

12 pages, 318 KB

Open AccessProceeding Paper

Abelian Groups of Fractional Operators

by Anthony Torres-Hernandez, Fernando Brambila-Paz and Rafael Ramirez-Melendez

Comput. Sci. Math. Forum 2022, 4(1), 4; https://doi.org/10.3390/cmsf2022004004 - 19 Dec 2022

Cited by 2 | Viewed by 4413

Abstract

Taking into count the large number of fractional operators that have been generated over the years, and considering that their number is unlikely to stop increasing at the time of writing this paper due to the recent boom of fractional calculus, everything seems to indicate that an alternative that allows to fully characterize some elements of fractional calculus is through the use of sets. Therefore, this paper presents a recapitulation of some fractional derivatives, fractional integrals, and local fractional operators that may be found in the literature, as well as a summary of how to define sets of fractional operators that allow to fully characterize some elements of fractional calculus, such as the Taylor series expansion of a scalar function in multi-index notation. In addition, it is presented a way to define finite and infinite Abelian groups of fractional operators through a family of sets of fractional operators and two different internal operations. Finally, using the above results, it is shown one way to define commutative and unitary rings of fractional operators. Full article

(This article belongs to the Proceedings of The 5th Mexican Workshop on Fractional Calculus)

11 pages, 247 KB

Open AccessArticle

Neutrosophic Components Semigroups and Multiset Neutrosophic Components Semigroups

by Vasantha W. B., Ilanthenral Kandasamy and Florentin Smarandache

Symmetry 2020, 12(5), 818; https://doi.org/10.3390/sym12050818 - 16 May 2020

Cited by 8 | Viewed by 2456

Abstract

Neutrosophic components (NC) under addition and product form different algebraic structures over different intervals. In this paper authors for the first time define the usual product and sum operations on NC. Here four different NC are defined using the four different intervals: (0, 1), [0, 1), (0, 1] and [0, 1]. In the neutrosophic components we assume the truth value or the false value or the indeterminate value to be from the intervals (0, 1) or [0, 1) or (0, 1] or [0, 1]. All the operations defined on these neutrosophic components on the four intervals are symmetric. In all the four cases the NC collection happens to be a semigroup under product. All of them are torsion free semigroups or weakly torsion free semigroups. The NC defined on the interval [0, 1) happens to be a group under addition modulo 1. Further it is proved the NC defined on the interval [0, 1) is an infinite commutative ring under addition modulo 1 and usual product with infinite number of zero divisors and the ring has no unit element. We define multiset NC semigroup using the four intervals. Finally, we define n-multiplicity multiset NC semigroup for finite n and these two structures are semigroups under + modulo 1 and

{M (S), +, \times}

and

{n

M (S), +, \times}

are NC multiset semirings. Several interesting properties are discussed about these structures. Full article

(This article belongs to the Special Issue New Development of Neutrosophic Probability, Neutrosophic Statistics, Neutrosophic Algebraic Structures, and Neutrosophic & Plithogenic Optimizations)

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