  Let $R$ be a Dedekind ring, $K$ its quotient field, and $L=K(\alpha)$ a
finite field extension of $K$ defined by a monic irreducible polynomial
$f(x)\in R[x]$. We give an easy version of Dedekind's criterion which
computationally improves those versions know in the literature. We further use
this result to give a sufficient condition for the integral closedness of
$R[\alpha]$ when $f(x)=x^n-a$. In case $R$ is a ring of integers of a number
field, we give yet sufficient and necessary conditions for this to hold,
generalizing and improving in both cases some known results in this direction.
Some highlighting examples are also given.
