We introduce the notion of nilpotent p.p. rings, and prove that the nilpotent p.p. condition is preserved over polynomial rings and skew polynomial rings.

Let A be a unital R-algebra and M be a unital A- bimodule. It is shown that every Jordan derivation of the trivial extension of A by M, under some conditions, is the sum of a derivation and an antiderivation.

A specialization semilattice is a join semilattice together with a coarser preorder ⊑,satisfying an appropriate compatibility condition. If 𝑋,is a topological space, then (P(𝑋, ), ∪, , ⊑, ) is a specialization semilattice, where 𝑥,⊑,𝑦,if 𝑥,⊆,𝐾, 𝑦, , for 𝑥, , 𝑦,⊆,𝑋, , and 𝐾,is closure. Specialization semilattices and posets appear as auxiliary structures in many disparate scientific fields, even unrelated to topology. In a former work we showed that every specialization semilattice can be embedded into the specialization semilattice associated to a topological space as above. Here we describe the universal embedding of a specialization semilattice into an additive closure semilattice.