Preserving Lefschetz properties after extension of variables

Consider a standard graded artinian $k$-algebra $B$ and an extension of $B$ by a new variable, $A=B{\otimes }_{k}k[x]\mathrm{/}(x^d)$ for some $d\ge 1$. We will show how maximal rank properties for powers of a general linear form on $A$ can be determined by maximal rank properties for different powers of general linear forms on $B$. This is then used to study Lefschetz properties of algebras that can be obtained via such extensions. In particular, it allows for a new proof that monomial complete intersections have the strong Lefschetz property over a field of characteristic zero. Moreover, it gives a recursive formula for the determinants that show up in that case. Finally, for algebras over a field of characteristic zero, we give a classification for what properties $B$ must have for all extensions $B{\otimes }_{k}k[x]\mathrm{/}(x^d)$ to have the weak or the strong Lefschetz property.