We define and completely explore the so-called WJU rings. This class properly encompasses the class of JU rings, introduced and studied by the present author in detail in Toyama Math. J. (2016).
"Weakly JU Rings." Missouri J. Math. Sci. 29 (2) 184 - 196, November 2017. https://doi.org/10.35834/mjms/1513306830