Note on the Davenport constant of the multiplicative semigroup of the quotient ring F-p[x]/< f(x)>

作者：Wang, Haoli^*; Zhang, Lizhen^*; Wang, Qinghong^*; Qu, Yongke^*

来源：International Journal of Number Theory, 2016, 12(3): 663-669.

DOI：10.1142/S1793042116500433

• Davenport constant
• zero-sum
• finite commutative semigroups
• polynomial rings

Let S be a finite commutative semigroup. The Davenport constant of S, denoted D(S), is defined to be the least positive integer l such that every sequence T of elements in S of length at least l contains a proper subsequence T' with the sum of all terms from T' equaling the sum of all terms from T. Let F-p[x] be a polynomial ring in one variable over the prime field F-p, and let f(x) is an element of F-p[x]. In this paper, we made a study of the Davenport constant of the multiplicative semigroup of the quotient ring F-p[x]/< f(x)> and proved that, for any prime p > 2 and any polynomial f(x) is an element of F-p[x] which factors into a product of pairwise non-associate irreducible polynomials,
D(S-f(x)(p)) = D(U(S-f(x)(p))),
where S-f(x)(p) denotes the multiplicative semigroup of the quotient ring F-p[x]/< f(x)> and U(S-f(x)(p)) denotes the group of units of the semigroup S-f(x)(p).

• 出版日期2016-5
• 单位天津理工大学; 天津工业大学; 上海大学; 天津师范大学; 洛阳师范学院