  In this paper, we study the notion of chordality and cycles in hypergraphs
from a commutative algebraic point of view. The corresponding concept of
chordality in commutative algebra is having a linear resolution. However, there
is no unified definition for cycle or chordality in hypergraphs in the
literature, so we consider several generalizations of these notions and study
their algebraic interpretations. In particular, we investigate the relationship
between chordality and having linear quotients in some classes of hypergraphs.
Also we show that if $\mathcal{C}$ is a hypergraph such that $\langle
\mathcal{C} \rangle$ is a vertex decomposable simplicial complex or
$I(\bar{\mathcal{C}})$ is squarefree stable, then $\mathcal{C}$ is chordal
according to one of the most promising definitions.
