Principal ideal domains and Euclidean domains having 1 as the only unit

We consider a question raised by Mowaffaq Hajja about the structure of a pr incipal ideal domain R having the property that 1 is the only unit of R. We also examine this unit condition for the case where R is a Euclidean domai n. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements F-2 or to the polynomial ring F-2[X] On the other hand, we establish existence of finitely generate d principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to F-2 or to F-2[X]. We also construct principal ideal domains R of infinite transcendence degree over F-2 with the property that 1 is the only unit of R.