# A Generalization of Reflexive Rings

## TL;DR

This paper introduces the concept of J-reflexive rings, a broad generalization of reflexive and J-reversible rings, providing characterizations, properties, and invariance results for this new class.

## Contribution

It defines J-reflexive rings, extends known results from reflexive rings to this new class, and shows the property is Morita invariant.

## Key findings

- J-reflexive rings generalize reflexive rings.
- Results of reflexive rings extend to J-reflexive rings.
- J-reflexive property is Morita invariant.

## Abstract

In this paper, we introduce a class of rings which is a generalization of reflexive rings and $J$-reversible rings. Let $R$ be a ring with identity and $J(R)$ denote the Jacobson radical of $R$. A ring $R$ is called {\it $J$-reflexive} if for any $a$, $b \in R$, $aRb = 0$ implies $bRa \subseteq J(R)$. We give some characterizations of a $J$-reflexive ring. We prove that some results of reflexive rings can be extended to $J$-reflexive rings for this general setting. We conclude some relations between $J$-reflexive rings and some related rings. We investigate some extensions of a ring which satisfies the $J$-reflexive property and we show that the $J$-reflexive property is Morita invariant.