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Non-commutative unique factorization rings. (English) Zbl 0601.16001
A 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
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