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.
ASJC Scopus subject areas
- Algebra and Number Theory