Note on the Davenport constant of the multiplicative semigroup of the quotient ring 𝔽p[x] 〈f(x)〉

2016 ◽  
Vol 12 (03) ◽  
pp. 663-669 ◽  
Author(s):  
Haoli Wang ◽  
Lizhen Zhang ◽  
Qinghong Wang ◽  
Yongke Qu
Keyword(s):  
Positive Integer ◽  
Commutative Semigroup ◽  
Polynomial Ring ◽  
Quotient Ring ◽  
Multiplicative Semigroup ◽  
Davenport Constant ◽  
Prime Field ◽  
Group Of Units ◽  
Polynomial Formula ◽  
Finite Commutative Semigroup

Let [Formula: see text] be a finite commutative semigroup. The Davenport constant of [Formula: see text], denoted [Formula: see text], is defined to be the least positive integer [Formula: see text] such that every sequence [Formula: see text] of elements in [Formula: see text] of length at least [Formula: see text] contains a proper subsequence [Formula: see text] with the sum of all terms from [Formula: see text] equaling the sum of all terms from [Formula: see text]. Let [Formula: see text] be a polynomial ring in one variable over the prime field [Formula: see text], and let [Formula: see text]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring [Formula: see text] and proved that, for any prime [Formula: see text] and any polynomial [Formula: see text] which factors into a product of pairwise non-associate irreducible polynomials, [Formula: see text] where [Formula: see text] denotes the multiplicative semigroup of the quotient ring [Formula: see text] and [Formula: see text] denotes the group of units of the semigroup [Formula: see text].

scholarly journals On a problem of Baayen and Kruyswijk

1968 ◽  
Vol 16 (2) ◽  
pp. 145-149
Author(s):  
D. A. Burgess
Keyword(s):  
Positive Integer ◽  
Commutative Semigroup ◽  
Finite Semigroup ◽  
Multiplicative Semigroup ◽  
Residue Classes ◽  
Finite Commutative Semigroup

We shall call a finite semigroup S arithmetical if there exists a positive integer N and a monomorphism μ of S into the multiplicative semigroup RN of the ring of residue classes of the integers modulo N. In 1965 P. C. Baayen and D. Kruyswijk [1] posed the problem' Is every finite commutative semigroup arithmetical? ' The purpose of this paper is to answer this question.

The Erdős–Burgess constant of the multiplicative semigroup of a factor ring of 𝔽q[x]

2019 ◽  
Vol 15 (01) ◽  
pp. 131-136 ◽  
Author(s):  
Haoli Wang ◽  
Jun Hao ◽  
Lizhen Zhang
Keyword(s):  
Lower Bound ◽  
Finite Field ◽  
Prime Ideal ◽  
Polynomial Ring ◽  
Prime Power ◽  
Quotient Ring ◽  
Prime Ideals ◽  
Multiplicative Semigroup ◽  
Associative Operation ◽  
Sharp Lower Bound

Let [Formula: see text] be a commutative semigroup endowed with a binary associative operation [Formula: see text]. An element [Formula: see text] of [Formula: see text] is said to be idempotent if [Formula: see text]. The Erdős–Burgess constant of [Formula: see text] is defined as the smallest [Formula: see text] such that any sequence [Formula: see text] of terms from [Formula: see text] and of length [Formula: see text] contains a nonempty subsequence, the sum of whose terms is idempotent. Let [Formula: see text] be a prime power, and let [Formula: see text] be the polynomial ring over the finite field [Formula: see text]. Let [Formula: see text] be a quotient ring of [Formula: see text] modulo any ideal [Formula: see text]. We gave a sharp lower bound of the Erdős–Burgess constant of the multiplicative semigroup of the ring [Formula: see text], in particular, we determined the Erdős–Burgess constant in the case when [Formula: see text] is the power of a prime ideal or a product of pairwise distinct prime ideals in [Formula: see text].

scholarly journals Elliptic curve over a finite ring generated by 1 and an idempotent element $e$ with coefficients in the finite field $\mathbb{F}_{3^d}$

2021 ◽  
Vol 40 ◽  
pp. 1-19
Author(s):  
A. Boulbot ◽  
Abdelhakim Chillali ◽  
A Mouhib
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Elliptic Curve ◽  
Polynomial Ring ◽  
Quotient Ring ◽  
Finite Ring ◽  
Idempotent Element ◽  
Characteristic 3 ◽  
Specific Equation ◽  
Weierstrass Equation

An elliptic curve over a ring $\mathcal{R}$ is a curve in the projective plane $\mathbb{P}^{2}(\mathcal{R})$ given by a specific equation of the form $f(X, Y, Z)=0$ named the Weierstrass equation, where $f(X, Y, Z)=Y^2Z+a_1XYZ+a_3YZ^2-X^3-a_2X^2Z-a_4XZ^2-a_6Z^3$ with coefficients $a_1, a_2, a_3, a_4, a_6$ in $\mathcal{R}$ and with an invertible discriminant in the ring $\mathcal{R}.$ %(see \cite[Chapter III, Section 1]{sil1}).  In this paper, we consider an elliptic curve over a finite ring of characteristic 3 given by the Weierstrass equation: $Y^2Z=X^3+aX^2Z+bZ^3$ where $a$ and $b$ are in the quotient ring $\mathcal{R}:=\mathbb{F}_{3^d}[X]/(X^2-X),$ where $d$ is a positive integer and $\mathbb{F}_{3^d}[X]$ is the polynomial ring with coefficients in the finite field $\mathbb{F}_{3^d}$ and such that $-a^3b$ is invertible in $\mathcal{R}$.

scholarly journals Commutative semigroups whose endomorphisms are power functions

2021 ◽  
Author(s):  
Ryszard Mazurek
Keyword(s):  
Positive Integer ◽  
Power Function ◽  
Cyclic Group ◽  
Commutative Semigroup ◽  
Commutative Monoid ◽  
Power Functions ◽  
Commutative Semigroups ◽  
Finite Cyclic Group

AbstractFor any commutative semigroup S and positive integer m the power function $$f: S \rightarrow S$$ f : S → S defined by $$f(x) = x^m$$ f ( x ) = x m is an endomorphism of S. We partly solve the Lesokhin–Oman problem of characterizing the commutative semigroups whose all endomorphisms are power functions. Namely, we prove that every endomorphism of a commutative monoid S is a power function if and only if S is a finite cyclic group, and that every endomorphism of a commutative ACCP-semigroup S with an idempotent is a power function if and only if S is a finite cyclic semigroup. Furthermore, we prove that every endomorphism of a nontrivial commutative atomic monoid S with 0, preserving 0 and 1, is a power function if and only if either S is a finite cyclic group with zero adjoined or S is a cyclic nilsemigroup with identity adjoined. We also prove that every endomorphism of a 2-generated commutative semigroup S without idempotents is a power function if and only if S is a subsemigroup of the infinite cyclic semigroup.

scholarly journals On Segre products of affine semigroup rings

1988 ◽  
Vol 110 ◽  
pp. 113-128 ◽  
Author(s):  
Lê Tuân Hoa
Keyword(s):  
Positive Integer ◽  
Polynomial Ring ◽  
Semigroup Ring ◽  
Semigroup Rings ◽  
Finitely Generated ◽  
Affine Semigroup

Let N denote the set of non-negative integers. An affine semigroup is a finitely generated submonoid S of the additive monoid Nm for some positive integer m. Let k[S] denote the semigroup ring of S over a field k. Then one can identify k[S] with the subring of a polynomial ring k[t1, …, tm] generated by the monomials .

A Class of Trinomials with Galois Group Sn

2012 ◽  
Vol 19 (spec01) ◽  
pp. 905-911 ◽  
Author(s):  
Anuj Bishnoi ◽  
Sudesh K. Khanduja
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Galois Group ◽  
Rational Numbers ◽  
Alternating Group ◽  
Polynomial Formula

A well known result of Schur states that if n is a positive integer and a0, a1,…,an are arbitrary integers with a0an coprime to n!, then the polynomial [Formula: see text] is irreducible over the field ℚ of rational numbers. In case each ai = 1, it is known that the Galois group of fn(x) over ℚ contains An, the alternating group on n letters. In this paper, we extend this result to a larger class of polynomials fn(x) which leads to the construction of trinomials of degree n for each n with Galois group Sn, the symmetric group on n letters.

On zero-sum subsequences of prescribed length

2017 ◽  
Vol 14 (01) ◽  
pp. 167-191 ◽  
Author(s):  
Dongchun Han ◽  
Hanbin Zhang
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Finite Abelian Group ◽  
Davenport Constant ◽  
Zero Sum

Let [Formula: see text] be an additive finite abelian group with exponent [Formula: see text]. For any positive integer [Formula: see text], let [Formula: see text] be the smallest positive integer [Formula: see text] such that every sequence [Formula: see text] in [Formula: see text] of length at least [Formula: see text] has a zero-sum subsequence of length [Formula: see text]. Let [Formula: see text] be the Davenport constant of [Formula: see text]. In this paper, we prove that if [Formula: see text] is a finite abelian [Formula: see text]-group with [Formula: see text] then [Formula: see text] for every [Formula: see text], which confirms a conjecture by Gao et al. recently, where [Formula: see text] is a prime.

scholarly journals On the monoid of cofinite partial isometries of $\qq{N}^n$ with the usual metric

2019 ◽  
Vol 12 (3) ◽  
pp. 51-68
Author(s):  
Oleg Gutik ◽  
Anatolii Savchuk
Keyword(s):  
Positive Integer ◽  
Symmetric Group ◽  
Natural Partial Order ◽  
Group Congruence ◽  
Partial Isometries ◽  
Group Of Units ◽  
Minimum Group ◽  
Positive Integers ◽  
Quotient Semigroup ◽  
Free Semilattice

In this paper we study the structure of the monoid Iℕn ∞ of  cofinite partial isometries of the n-th power of the set of positive integers ℕ with the usual metric for a positive integer n > 2. We describe the group of units and the subset of idempotents of the semigroup Iℕn ∞, the natural partial order and Green's relations on Iℕn ∞. In particular we show that the quotient semigroup Iℕn ∞/Cmg, where Cmg is the minimum group congruence on Iℕn ∞, is isomorphic to the symmetric group Sn and D = J in Iℕn ∞. Also, we prove that for any integer n ≥2 the semigroup Iℕn ∞  is isomorphic to the semidirect product Sn ×h(P∞(Nn); U) of the free semilattice with the unit (P∞(Nn); U)  by the symmetric group Sn.

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