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

Abstract

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].