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Inspired by recent work of Carlson, Friedlander and Pevtsova concerning modules for  p -elementary abelian groupsE  r{E_{r}}    of rank  r  over a field of characteristicp  &gt;  0{p&gt;0}   , we introduce the notions of modules with constant  d -radical rank and modules with constant  d -socle rank for the generalized Kronecker algebra  r   =   k  ⁢   Γ  r{\mathcal{K}_{r}=k\Gamma_{r}}    withr  ≥  2{r\geq 2}    arrows and1  ≤  d  ≤   r  -  1{1\leq d\leq r-1}   . We study subcategories given by modules with the equal  d -radical property and the equal  d -socle property. Utilizing the simplification method due to Ringel, we prove that these subcategories inmod  ⁡     r{\operatorname{mod}\mathcal{K}_{r}}    are of wild type. Then we use a natural functor  :    mod  ⁡     r    →   mod  ⁡   k  ⁢   E  r{\operatorname{\mathfrak{F}}\colon{\operatorname{mod}\mathcal{K}_{r}}\to% \operatorname{mod}kE_{r}}    to transfer our results tomod  ⁡   k  ⁢   E  r{\operatorname{mod}kE_{r}}   .

Representations of constant socle rank for the Kronecker algebra