The notion of reflexivity is one of the very well-known notions of ring theory, which was introduced by Meson [Comm. Algebra, 9, 1709-1724 (1981)]. He called a ring R reflexive if aRb=0 implies bRa=0 for any a, b ∈ R. During the period, many ring theorists studied and generalized the notion of reflexive rings. In 2015, Chakraborty [Asian-European J. Math., 8, 1550003:1-15, (2015)] introduced the notion of central reflexive rings as a generalization of reflexive rings. He called a ring R central reflexive if aRb = 0 implies bRa ⊆ C(R) for any a, b ∈ R, where C(R) denotes the set of all central elements of R. Recently, Calci et al. [Math. Bohemica, 149, 225-235 (2024)] introduced the notion of J-reflexive rings as a generalization of central reflexive rings. He called a ring R J-reflexive if aRb = 0 implies bRa ⊆ J(R) for any a, b ∈ R, where J(R) denotes the Jacobson radical of R.

Here, we introduce the notion of H-reflexive rings, which lies strictly between the classes of central reflexive rings and J-reflexive rings. In support, we give some examples and counterexamples. We find that the notions of reflexive, central reflexive, H-reflexive, and J-reflexive rings are equivalent to the class of J-semisimple rings; and the notions of H-reflexive and central reflexive rings are equivalent to the class of rings with no nil ideals. We prove that a ring R is H-reflexive if and only if, for every a, b ∈ R, < a >< b >= 0 implies < b >< a > ⊆ T(R), where T(R) denotes the hypercenter of R. We also give some characterizations of H-reflexive rings with the help of their extension rings. Finally, we discuss some of their basic properties and their relationship with the class of Armendariz rings.