Note on the Davenport constant of the multiplicative semigroup of the quotient ring \(\frac{\mathbb{F}_p[x]}{\langle f(x)\rangle}\). (Q2800150)

The Davenport constant \(D(S)\) of a commutative semigroup written additively has been defined by \textit{G. Wang} and \textit{W. Gao} [Semigroup Forum 76, No. 2, 234-238 (2008; Zbl 1142.20043)] as the smallest integer \(D>0\) such that every sequence \(T\) of elements of \(S\) having \(\geq D\) elements has a proper subsequence \(T'\) with \(\sum_{a\in T'}a=\sum_{a\in T}a\).NEWLINENEWLINE The authors consider the semigroup \(S=\prod_{i=1}^kS_i\), where \(S_i\) is the monoid obtained by adjoining to the cyclic group \(C_{n_i}\) an element \(\infty_i\) satisfying \(a+\infty_i=\infty_i\). They show in Lemma 2.3 that \(D(S)=D(\prod_{i=1}^kC_{n_i})\), and apply this to the case when \(S\) is the multiplicative semigroup of the ring \(F_p[X]/f(X)F_p[X]\), where \(f(X)\) is a product of non-associated irreducible polynomials.