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 principal 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 domain. We prove that a finitely generated Euclidean domain having 1 as its only unit is isomorphic to the field with two elements double-struck F sign₂ or to the polynomial ring double-struck F sign₂[X]. On the other hand, we establish existence of finitely generated principal ideal domains R such that 1 is the only unit of R and R is not isomorphic to double-struck F sign₂ or to double-struck F sign₂[X]. We also construct principal ideal domains R of infinite.