This paper studies effective aspects of semiperfect rings from the standpoint of reverse mathematics. Based on first-order Jacobson radicals of rings, we define a ring R with the Jacobson radical Jac(R) to be semiperfect if the quotient ring R/Jac(R) is semisimple, and idempotents of the quotient ring can be lifted to R. Using elementary matrix operations in linear algebra, we show that RCA₀ proves a characterization of semiperfect rings in terms of idempotents of rings. Semiperfect rings are generalizations of semisimple rings and local rings, and semiperfect rings R with R/Jac(R) simple are isomorphic to matrix rings over local rings. Based on the effective characterization of semiperfect rings via idempotents, we prove the structure theorem of semiperfect rings R with R/Jac(R) simple in RCA₀. Left perfect rings or right perfect rings are always semiperfect. Finally, we provide a proof for the structure theorem of one-sided perfect rings R with R/Jac(R) simple in WKL₀.

In [S. Celani and L. Cabrer. Duality for finite Hilbert algebras. Discrete Math., 305(1-3):74{99, 2005.] the authors proved that every finite Hilbert algebra A is isomorphic to the Hilbert algebra H_K(X) = {w ⇒ _i v : w ∈ K and v ⊆ w}, where X is a finite poset, K is a distinguished collection of subsets of X, and the implication ⇒_i is defined by: w ⇒_i  v = {x ∈ X : w ∩ [x) ⊆ v}, where [x) = {y ∈ X : x ≤ y.} The Hilbert implication on H_K(X) is the usual Heyting implication ⇒ _i  (as just defined) given on the increasing subsets. In the same article, Celani and Cabrer extended this representation to a full categorical duality. The aim of the present article is to obtain an algebraic characterization of the Hilbert algebras H_K(X) for all structures (X, ≤, K) defined by Celani and Cabrer but not necessarily finite. Then, we shall extend this  representation to a full dual equivalence generalizing the finite setting given by Celani and Cabrer.

We show that a variety with J´onsson terms t₁, . . . , t_n–1 has directed J´onsson terms d₁, . . . , d_n–1 for the same value of the indices, solving a problem raised by Kazda et al. Refined results are obtained for locally finite varieties.