Semigroup rings and algebraically independent sequences with respect to idempotents in commutative semigroups

Abstract

For any finite abelian group $G$ and commutative unitary ring $R$, by $R[G]$ we denote the group algebra over $R$. Let $T=(g_1,\ldots,g_{\ell})$ be a sequence over the group $G$. We say $T$ is algebraically zero-sum free over R if $\prod\limits_{i=1}^r(X^{g_i}-a_i)\neq 0\in R[G]$ for all $a_1,\ldots,a_{\ell}\in R\setminus \{0\}$. Let $d(G,R)={\rm sup}\{|T|: T \mbox{ is an algebraically zero-sum free sequence over } R \mbox{ of terms from }G\}.$ This invariant of the group algebra $R[G]$ plays a powerful role in the research for the zero-sum theory. In this paper, we generalize this invariant to the semigroup algebra $R[S]$ for a commutative periodic semigroup $S$. We give the best possible lower and upper bounds for $d(S,R)$ for a general commutative periodic semigroup $S$. In case that $K$ is a field, and $S$ is a finite commutative semigroup, we give more precise result, including the equality for Clifford semigroups, Archimedean semigroups and elementary semigroups, which covers all types of irreducible components associated with the semilattice decomposition and the subdirect product decomposition of a commutative semigroup. Also, the invariant $d(S,K)$ was applied to the study of some zero-sum invariants in semigroups. One conjecture on the equality for $d(S,K)$ in case $K$ is an algebraically closed field of characteristic zero was proposed which has been also partially affirmed in this paper.