Summary: For a commutative ring \(R\) with identity, let \(\langle a\rangle\) be the principal ideal generated by \(a\in R\). Let \(\Omega (R)^*\) be the set of all nonzero proper principal ideals of \(R\). The reduced cozero-divisor graph \(\Gamma_r (R)\) of \(R\) is the simple undirected graph whose vertex set is \(\Omega (R)^*\) and such that two distinct vertices \(\langle a\rangle\) and \(\langle b\rangle\) in \(\Omega (R)^{\ast}\) are adjacent if and only if \(\langle a \rangle\nsubseteq\langle b\rangle\) and \(\langle b\rangle\nsubseteq\langle a\rangle\). In this article, we study certain properties of embeddings of the reduced cozero-divisor graph of commutative rings. More specifically, we characterize all Artinian nonlocal rings whose reduced cozero-divisor graph has genus two. Also we find the book thickness of the reduced cozero-divisor graphs which have genus at most one.