Strongly UR-Clean ring: Properties and relation to strongly clean ring

A ring R is called a strongly ur-clean ring, if every element x ∈ R can be expressed as x = a + e and ea = ae for a unit-regular element a and an idempotent element e. This article presents several properties of a strongly ur-clean ring, namely, if x is an element in a strongly ur-clean ring R, then the element (1 - x) ∈ R, properties of a strongly ur-clean ring on its factor ring, the eRe ring, the elemental characteristics of the Jacobson radical with its elements having special properties for a strongly ur-clean ring, and the unit element characteristics with special properties for a reduced strongly ur-clean ring. In addition, this article also presents the relationship between a strongly ur-clean ring and a strongly clean ring, as well as the properties of the elements of a strongly ur-clean ring to a strongly clean ring.