Quotients of preradicals over local uniserial rings

The lattice of preradicals over a local uniserial ring has been studied some years ago. In this paper, we study the lattice of quotients of preradicals , which were defined by M.L. Teply and J.E. van den Berg for any associative ring R with identity. When R is a local uniserial ring, we describe these quotients as ternary sequences of length n (the composition length of R ) and also as w -paths, which are defined here. These one-to-one correspondences allow us to see the set "Equation missing" of quotients of preradicals as a finite distributive lattice with some properties, and whose cardinality we obtain in terms of Motzkin numbers. We explore quotients of certain types of preradicals. We also present a monoid embedding from "Equation missing" to "Equation missing" , where R and $$R'$$ R ′ have, respectively, composition length n and $$n+1$$ n + 1 .