## 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 double-struck F sign_{2} or to the polynomial ring double-struck F sign_{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 double-struck F sign_{2} or to double-struck F sign_{2}[X]. We also construct principal ideal domains R of infinite.

Original language | English |
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Pages (from-to) | 5197-5208 |

Number of pages | 12 |

Journal | Communications in Algebra |

Volume | 29 |

Issue number | 11 |

DOIs | |

State | Published - 2001 |

## ASJC Scopus subject areas

- Algebra and Number Theory