A class of Hilbert series and the strong Lefschetz property

We determine the class of Hilbert series H H so that if M M is a finitely generated zero-dimensional R R -graded module with the strong Lefschetz property, then M ⊗ k k [ y ] / ( y m ) M⊗ _k k[y]/(y^m) has the strong Lefschetz property for an indeterminate y y and all positive integers m m if and only if the Hilbert series of M M is in H H . Given two finite graded R R -modules M M and N N with the strong Lefschetz property, we determine sufficient conditions in order that M ⊗ k N M⊗ _kN has the strong Lefschetz property.