  The study of the edge ideal $I(D_{G})$ of a weighted oriented graph $D_{G}$
with underlying graph $G$ started in the context of Reed-Muller type codes. We
generalize a Cohen-Macaulay construction for $I(D_{G})$, which Villarreal gave
for edge ideals of simple graphs. We use this construction to classify all the
Cohen-Macaulay weighted oriented edge ideals, whose underlying graph is a
cycle. We show that the conjecture on Cohen-Macaulayness of $I(D_{G})$,
proposed by Pitones et al. (2019), holds for $I(D_{C_{n}})$, where $C_{n}$
denotes the cycle of length $n$. Miller generalized the concept of Alexander
dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in
that direction, we study the Alexander dual of $I(D_{G})$ and its conditions to
be Cohen-Macaulay.
