The aim of the paper is to characterize the equation 1 + Δ(R) = U(R), where J(R) ⊆ Δ(R) = {x ∈ R : x + u ∈ U(R) for all u ∈ U(R)} that is the largest Jacobson radical subring of R and U(R) is the set of invertible elements of a ring R. We show that this equation is closely related to UJ-rings and rings whose elements can be written as the sum of an idempotent and an element from Δ(R). After presenting several characterizations and properties of this equation, we consider the rings satisfying the equation 1 + Δ(R) = U(R) within many well-studied classes of rings. Finally, we close the paper with group rings.

中文翻译：

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UJ环的概括

本文的目的是描述方程1 + Δ(R) = ü(R)， 在哪里Ĵ(R) ⊆ Δ(R) = {X ∈ R ： X + 你 ∈ ü(R) 对所有人 你 ∈ ü(R)}这是最大的 Jacobson 激进子环R和ü(R)是环的可逆元素的集合R. 我们证明这个方程与üĴ-环和环，其元素可以写为幂等和元素之和Δ(R). 在介绍了这个方程的几个特征和性质之后，我们考虑满足方程的环1 + Δ(R) = ü(R)在许多经过充分研究的环类中。最后，我们用群环关闭论文。