Non-commutative unique factorization rings.

*(English)*Zbl 0601.16001A prime (right and left) Noetherian ring R is called a UFR (unique factorization ring) if every non-zero prime ideal in R contains a non- zero principal prime ideal \(pR=Rp\). This is a generalization of an earlier definition given by the first author [in Math. Proc. Camb. Philos. Soc. 95, 49-54 (1984; Zbl 0541.16001)] where domains were investigated for which height-1 prime ideals were not only principal but also completely prime. It is pointed out that this last property is not stable under polynomial extensions. Here it is shown that R[x] is a UFR if R is a UFR. The twisted polynomial ring R[x,\(\alpha\) ], \(\alpha\) an automorphism of R, is a UFR if and only if R is a prime Noetherian ring and every non-zero \(\alpha\)-prime ideal of R contains a non-zero \(\alpha\)-prime ideal which is principal. A sufficient condition is given for R[x,\(\delta\) ], \(\delta\) a derivation, to be a UFR. One of the general results about UFR’s used to obtain the above results states that a UFR is the intersection of a simple partial quotient ring and the localizations at the height-1 primes. Examples are given and it follows that the universal enveloping algebra of a finite dimensional, solvable complex Lie algebra is a UFR.

Reviewer: H.-H.Brungs

##### MSC:

16U10 | Integral domains (associative rings and algebras) |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

16P40 | Noetherian rings and modules (associative rings and algebras) |

16P50 | Localization and associative Noetherian rings |

16N60 | Prime and semiprime associative rings |

17B35 | Universal enveloping (super)algebras |

16Dxx | Modules, bimodules and ideals in associative algebras |