Given a finite commutative semigroup $\mathcal{ S}$ (written additively), denoted by ${ \rm D}(\mathcal{ S})$ the Davenport constant of $\mathcal{ S}$, namely the least positive integer $\ell$ such that for any $\ell$ elements $s_1,\ldots,s_{ \ell}\in \mathcal{ S}$ there exists a set $I\subsetneq [1,\ell]$ for which $\sum_{ i\in I} s_i=\sum_{ i=1}^{ \ell} s_i$. Then, for any integers $r\geq 1, n_1,\ldots,n_r>1$, let $R=\mathbb{ Z}_{ n_1}\oplus\cdots\oplus \mathbb{ Z}_{ n_r}$ be the direct sum of these $r$ residue class rings $\mathbb{ Z}_{ n_1}, \ldots,\mathbb{ Z}_{ n_r}$. Moreover, let $\mathcal{ S}_R$ be the multiplicative semigroup of the ring $R$, and ${ \rm U}(\mathcal{ S}_R)$ the group of units of $\mathcal{ S}_R$. In this paper, we prove that $${ \rm D}({ \rm U}(\mathcal{ S}_R))+P_2\leq { \rm D}(\mathcal{ S}_R)\leq { \rm D}({ \rm U}(\mathcal{ S}_R))+\delta,$$ where $P_2=\sharp\{ i\in [1,r]: 2 \parallel n_i\}$ and $\delta=\sharp\{ i\in [1,r]: 2\mid n_i\}.$ This corrects our previous published wrong result on this problem.
Categories: math.CO math.NT