Davenport constant for commutative rings (Q344143)

Let \((R,\cdot,+)\) be a commutative ring. Then \((R,\cdot)\) is a commutative monoid. This paper relates the Davenport constant of this monoid and the Davenport constant of the group of units of \(R\).NEWLINENEWLINELet \(S\) be a commutative semigroup. Let \(d(S)\) be the smallest nonnegative integer (or infinity) such that for any positive integer \(m\) and \(s_1,\ldots,s_m\in S\) there exits \(J\subseteq \{1,\ldots, m\}\) with cardinality less than or equal to \(d\) such that \(\sum_{i=1}^m s_i = \sum_{j\in J}s_j\).NEWLINENEWLINEA sequence \(A\) of elements in \(S\) is said to be reducible if there is no proper subsequence \(B\) of \(A\) such that the sum of the elements in \(A\) equals the sum of the elements in \(B\). If this is not the case, then \(A\) is said to be irreducible.NEWLINENEWLINEThe Davenport constant of the commutative semigroup \(S\), denoted \(D(S)\), is the smallest \(D\in \mathbb{N}\cup\{\infty\}\) such that every sequence of \(D\) elements in \(S\) is reducible. It turns out that \(D(S)=d(S)+1\). This definition of Davenport constant extends naturally the classical definition for Abelian groups.NEWLINENEWLINEIt is known that if \(S\) has finitely many elements, then \(d(S)\), and thus \(D(S)\), are finite.NEWLINENEWLINEIf \(R\) is a commutative ring, and \(S_R\) is the underlying monoid \((R,\cdot)\), then the Davenport constant of \(R\) is defined as \(D(R)=D(S_R)\). The author proves that if \(D(R)\) is finite, then \(R\) has finitely many elements. So for rings, \(D(R)\) has finitely many elements if and only if \(D(R)\) is finite.NEWLINENEWLINEDue to this fact, the author focuses on finite commutative rings. Such rings can be decomposed as \((\mathbb{Z}_2)^{k_1}\times (\mathbb{Z}_4)^{k_2}\times (\mathbb{Z}_8)^{k_3}\times (\mathbb{F}_2[x]/(x^2))^{k_4}\times R'\), with \(R'\) not isomorphic to \(\mathbb{Z}_2\), \(\mathbb{Z}_4\), \(\mathbb{Z}_8\), \(\mathbb{F}_2[x]/(x^2)\). A formula of \(D(R)\) is given in terms of the Davenport constant of \(U(R')\) times a power of \(C_2\) depending on the constants \(k_1,\ldots, k_4\), with \(U(R')\) the group of units of \(R'\). This yields some interesting corollaries relating the Davenport constants of \(R\) and \(U(R)\).NEWLINENEWLINETo achieve this results, the author carefully studies the behavior under Cartesian product of the Davenport constant.