The purpose of this note is to communicate an error in [Reference Devhare, Joshi and LaGrange2] and correct the results that are affected. The error lies in the proof of [Reference Devhare, Joshi and LaGrange2, Lemma 3.8], where it was not taken into account that edges can appear between vertices of $G^c(A)$ when passing to $G^c(A\cup \{a\})$ . Consequently, it can happen that $F(x)=F(y)$ for $x,y\in V(G^c(A))$ with $\{x,y\}^\wedge \neq \{0\}$ , and there are three results, that is, [Reference Devhare, Joshi and LaGrange2, Theorem 1.1, Lemma 3.8 and Corollary 4.3], that require modification. In this note, we correct these results by introducing a condition on the set $\mathscr {A}$ of atoms of P (although, Lemma 3.8 is not addressed since it can be omitted under the new assumptions). By using the correction to [Reference Devhare, Joshi and LaGrange2, Theorem 1.1], we confirm that [Reference Devhare, Joshi and LaGrange2, Corollaries 4.1 and 4.2] do not require revision.

The following definitions will be used and all other notation is the same as in [Reference Devhare, Joshi and LaGrange2]. Let P be a partially ordered set with zero and $Z(P)\neq \{0\}$ . For $x\in P$ , we define $\textbf {a}(x)=\{a\in \mathscr {A} \mid a\leq x\}$ and let $\mathcal {C}_x=\{y\in P \mid \textbf {a}(y)=\mathscr {A}\setminus \textbf {a}(x)\}$ . Let $[x]=\{y\in P \mid \textbf {a}(x)=\textbf {a}(y)\}$ . If $\{\textbf {a}(x) \mid x\in P\}\cup \{\mathscr {A}\}$ is assumed to be a Boolean algebra (under inclusion), then $\mathcal {C}_x\neq \emptyset $ for every $x\in V(G^c(P))$ and the sets $[x]$ ( $x\in V(G^c(P))$ ) and $\mathscr {A}$ are finite if $\omega (G^c(P))<\infty $ . (For example, if $|\mathscr {A}|\geq 3$ , then $\bigcup _{a\in \mathscr {A}}\mathcal {C}_a$ induces a clique.) These facts will be freely applied throughout.

The following result serves as a correction to the statement of [Reference Devhare, Joshi and LaGrange2, Theorem 1.1].

The following arguments show that [Reference Devhare, Joshi and LaGrange2, Corollaries 4.1 and 4.2] remain valid and that [Reference Devhare, Joshi and LaGrange2, Corollary 4.3] holds for ‘distributive modules’.

Let R be a reduced commutative ring. Consider the partial order on R such that $r\leq s$ if and only if either $\text {ann}(s)\subsetneq \text {ann}(r)$ , or r is less than or equal to s in some fixed well-order on $\text {ann}(s)=\text {ann}(r)$ . It can be shown that $\Gamma (R)$ is equal to the zero-divisor graph $G(R)$ of $(R,\leq )$ [Reference LaGrange and Roy3, Remark 3.4]. One can check that if $r\in R$ and a is a (poset-theoretic) atom of R, then $a\leq r$ if and only if $ar\neq 0$ . It follows that if $a_1,\ldots ,a_k$ are atoms, then $\textbf {a}(a_1+\cdots +a_k)=\{a_1,\ldots ,a_k\}$ (for example, if $a_1\not \leq a_1+\cdots +a_k$ , then $a_1^2=a_1(a_1+\cdots +a_k)=0$ , which is a contradiction). Hence, if $\omega (G^c(R))<\infty $ , then $\{\textbf {a}(r) \mid r\in R\}$ is the Boolean algebra of all subsets of $\mathscr {A}$ . Similarly, in the lattice of ideals of R, if $\omega (\mathbb {AG}^c(R))<\infty $ , then $\{\textbf {a}(I) \mid I$ is an ideal of $R\}$ is a Boolean algebra since if $I_1,\ldots ,I_k$ are minimal ideals, then $\textbf {a}(I_1+\cdots +I_k)=\{I_1,\ldots ,I_k\}$ . To see this, if $I,I_1,\ldots ,I_k$ are minimal ideals and $I\not \in \{I_1,\ldots ,I_k\}$ , then $I\cap I_j=\{0\}$ implies $II_j=\{0\}$ , so $I(I_1+\cdots +I_k)=II_1+\cdots +II_k=\{0\}$ , and thus $I\cap (I_1+\cdots +I_k)=\{0\}$ (alternatively, the claim is easily checked by noting that the condition ‘ $\omega (\mathbb {AG}^c(R))<\infty $ ’ implies that R is a reduced Artinian ring [Reference Behboodi and Rakeei1, Theorem 1.1], that is, R is a finite direct product of fields).

Now, [Reference Devhare, Joshi and LaGrange2, Corollary 4.2] remains valid since, as noted above, the condition ‘ $\omega (\mathbb {AG}^c(R))<\infty $ ’ guarantees that R is reduced and Artinian, which implies every ideal is a sum of minimal ideals, and hence $|[I]|=1$ for every ideal I of R. To see that [Reference Devhare, Joshi and LaGrange2, Corollary 4.1] remains valid, note that if $\textbf {a}(x)=\{a_1,\ldots ,a_k\}$ and $\textbf {a}(y)=\{a_1,\ldots ,a_k,a_{k+1}\}$ , then $[x]\rightarrow [y]$ by $\alpha \mapsto \alpha +a_{k+1}$ is injective. (The property ‘ $a\leq r$ if and only if $ar\neq 0$ ’ mentioned above can be used to verify $\alpha +a_{k+1}\in [y]$ .)

However, [Reference Devhare, Joshi and LaGrange2, Corollary 4.3] needs revision. For this, observe that the argument in [Reference Devhare, Joshi and LaGrange2] given prior to Corollary 4.3 shows that if $(F\cup \{\emptyset \},\cap )$ is a meet-semilattice with $\omega (\mathbb {I}(F))<\infty $ , then $\mathbb {I}(F)$ is weakly perfect if $G^c(F\cup \{\emptyset \})$ is weakly perfect. Hence, to show that the intersection graph of an R-module M is weakly perfect, it is enough to show that the subgraph of $IG(M)$ induced by the nonessential submodules of M is weakly perfect.

An R-module M is called distributive if $N_1\cap (N_2+N_3)=(N_1\cap N_2)+(N_1\cap N_3)$ for all submodules $N_1,N_2,N_3\leq M$ (for example, this is the case for every cyclic group). If M is distributive with $\omega (IG(M))<\infty $ , then clearly $\{\textbf {a}(N) \mid N$ is a submodule of $M\}$ is the Boolean algebra of all subsets of minimal submodules (for example, similar to the argument given above for ideals, if $N,N_1,\ldots ,N_k$ are minimal submodules and $N\not \in \{N_1,\ldots ,N_k\}$ , then $N\cap (N_1+\cdots +N_k)=N\cap N_1+\cdots +N\cap N_k=\{0\}$ ). Also, if S and T are nonessential submodules of M with $\textbf {a}(S)=\{N_1,\ldots ,N_k\}$ and ${\textbf {a}(T)=\{N_1,\ldots ,N_k,N_{k+1}\}}$ , then $|[S]|\leq |[T]|$ since $[S]\rightarrow [T]$ by $N\mapsto N+N_{k+1}$ is injective; for example, if $N+N_{k+1}=N'+N_{k+1}$ , then

$$ \begin{align*}N=N\cap(N'+N_{k+1})=N\cap N'+N\cap N_{k+1}=N\cap N',\end{align*} $$

that is, $N\subseteq N'$ , and the reverse inclusion holds symmetrically.

By Theorem 1, this verifies the following correction to [Reference Devhare, Joshi and LaGrange2, Corollary 4.3].