  We introduce the concept of quasi-linearity and prove it is necessary for a
monomial ideal to have a linear resolution and identify all the quasi-linear
quadratic monomial ideals. We define a strongly linear monomial for a monomial
ideal $I$ and prove that if $\mathbf{u}$ is a strongly linear monomial over $I$
then $I$ has a linear resolution (resp: is quasi-linear) if and only if
$I+\mathbf{u}\mathfrak{p}$ has a linear resolution (resp: is quasi-linear).
Here $\mathfrak{p}$ is any monomial prime ideal.
