Linear resolutions and quasi-linearity of monomial ideals

We introduce the notion of quasi-linearity and prove it is necessary for a monomial ideal to have a linear resolution and clarify all the quasi-linear monomial ideals generated in degree $2$. We also introduce the notion of a strongly linear monomial over a monomial ideal and prove that if $\mathbf {u}$ is a monomial strongly linear over $I$ then $I$ has a linear resolution (respectively is quasi-linear) if and only if $I+\mathbf {u}\mathfrak {p}$ has a linear resolution (respectively is quasi-linear). Here $\mathfrak {p}$ is any monomial prime ideal.