PRINCIPAL IDEAL DOMAINS AND EUCLIDEAN DOMAINS HAVING 1 AS THE ONLY UNIT

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.