Distributive factor lattices in free rings

Abstract:

For any field $E$ with subfield $k$ the free $E$-ring over $k$ on a set $X,\quad R = {\text { }}{E_k}\left \langle X \right \rangle$ is a fir. It is proved here that when $E/k$ is purely inseparable, then the submodule lattice $R/cR$ is distributive, for any $c \ne 0$ ( $R$ has distributive factor lattice); by contrast this is false when $E/k$ is a nontrivial Galois extension and $X \ne \emptyset$.