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Slicing a module into semisimple ones is useful to study modules. Loewystructures provide a means of doing so. To establish the Loewy structures ofprojective modules over a finite dimensional symmetric algebra over a field$F$, the Landrock lemma is a primary tool. The lemma and its corollary relateradical layers of projective indecomposable modules to radical layers of the$F$-duals of those modules ("dual symmetry") and to socle layers of thosemodules ("reciprocity").  We generalize these results to an arbitrary finite dimensional algebra $A$.Our main theorem, which is the same as the Landrock lemma for finitedimensional symmetric algebras, relates radical layers of projectiveindecomposable modules $P$ to radical layers of the $A$-duals of those modulesand to socle layers of injective indecomposable modules $\nu P$ where$\nu({-})$ is the Nakayama functor. A key tool to prove the main theorem is apair of adjoint functors, which we call socle functors and capital functors.

A generalization of dual symmetry and reciprocity for symmetric algebras