Rings such that, for each unit u, u − u^n belongs to the Jacobson radical

The focus of this article is the rings R for which u−u^n belongs to the Jacobson radical for all units u of R, where n ≥ 1 is a fixed integer. Such rings are called n-UJ rings. The structures of these rings are known if n = 1 or 2. As well as, we study the n-UJ property under some algebraic construction, the trivial extension and the Morita context of n-UJ rings are obtained.