Abstract
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 F 2 or to the polynomial ring F 2[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 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.
*Part of this work was prepared while M. Roitman enjoyed the hospitality of Purdue University.
Acknowledgments
Notes
*Part of this work was prepared while M. Roitman enjoyed the hospitality of Purdue University.