so $R/I$ has the weak Lefschetz property if $\operatorname{char}{K} \geq p$.

And (pseudo-)conversely, again for monomial ideals, if the weak Lefschetz property holds in some positive characteristic, then it holds for characteristic zero.

\[lem:wlp-p-zero\] Let $I$ be an artinian monomial ideal in $R$. If $R/I$ has the weak Lefschetz property when $\operatorname{char}{K} = p > 0$, then $R/I$ has the weak Lefschetz property for $\operatorname{char}{K} = 0$.

The proof is the same as that of Lemma \[lem:wlp-ae\] except we notice that if an integer $d$ is non-zero modulo a prime $p$, then $d$ is not zero.

Last, we note that any artinian ideal in two variables has the weak Lefschetz property. This was proven for characteristic zero in [@HMNW Proposition 4.4] and then for arbitrary characteristic in [@MZ Corollary 7], though it was not specifically stated therein, as noted in [@LZ Remark 2.6]. We provide a brief, direct proof of this fact to illustrate the weak Lefschetz property. Unfortunately, the simplicity of this proof fails in three variables, even for monomial ideals.

\[pro:2-wlp\] Let $R = K[x,y]$, where $K$ is an infinite field with [*arbitrary*]{} characteristic. Every artinian algebra in $R$ has the weak Lefschetz property.

Assume $I = (g_1, 