Abstract:

We define and explore a new class of rings called unit-exchange rings that strictly contain the class of exchange rings. Recall that a ring R is exchange if for each element a ∈ R there exists an idempotent e ∈ R such e ∈ aR and (1−e) ∈ (1−a)R. We have visualized unit-exchange to be a property such that for each a ∈ R there exists a unit u such that the element au satisfies the exchange property. A ring R is called left unit-exchange if for each element a ∈ R, there exists a unit u and an idempotent e in R such that e − ua ∈ R(a − aua). Similarly we can define the ring to be right unit-exchange if for some unit v and an idempotent f we have f − av ∈ (a − ava)R. We will show that the definition of unit-exchange is left-right symmetric. We will give examples of several classes of unit-exchange rings, some of which are not exchange. We will show that if R is unit-exchange then Mn(R) is also unit-exchange. It can be easily seen that any left (right) uniquely generated ring is always directly finite. We will show that the above conditions are equivalent in a unit-exchange ring.