Davenport constant of the multiplicative semigroup of the ring ${\mathbb{Z}}_{n_1}\oplus \cdots \oplus {\mathbb{Z}}_{n_r}$

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