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If $R$ is a commutative, unital ring, then $R \cong_{\sf Ring} {\rm End}_{{\sf Mod}_R}(R_R)$, with the elements of $R$ acting on $R$ by left multiplication. Now let $R$ be any non-atomic integral domain and note that the multiplicative monoid of ${\rm End}_{{\sf Mod}_R}(R_R)$ is a divisor-closed submonoid of the multiplicative monoid of ${\rm End}_{{\sf Grp}}(R_R)$. Lastly, recall that a domain is atomic (i.e., every non-unit is a product of some irreducible elements) iff so is its multiplicative monoid, and that a monoid $H$ is atomic only if so are all the divisor-closed submonoids of $H$ (we say that a submonoid $M$ of $H$ is divisor-closed if $x \mid_H y$ and $y \in M$ imply $x \in M$).