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**Non-commutative unique factorization domains.**
*(English)*
Zbl 0541.16001

A class of rings is introduced that includes commutative unique factorization domains as well as enveloping algebras of solvable Lie algebras. The definition is as follows: A ring R is called a U.F.D. if R is a Noetherian integral domain such that every height one prime P in R is completely prime and of the form \(P=pR=Rp\) for some element p in R (called prime). It is further assumed that at least one height one prime exists. The elements a in R can then be written in the form \(a=cp_ 1...p_ n\) for primes \(p_ i\) in R and an element c in R not contained in any height one prime P. Let C be the set of these elements c. The quotient ring \(RC^{-1}=T\) of R with respect to C exists, is a U.F.D. in which all one sided ideals are two-sided and is a maximal order in the skew field of quotients of R. Various examples are given.

Reviewer: H.-H.Brungs

### MSC:

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

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

16P50 | Localization and associative Noetherian rings |

17B35 | Universal enveloping (super)algebras |

16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |

### Keywords:

unique factorization domains; enveloping algebras of solvable Lie algebras; Noetherian integral domain; height one prime; completely prime; maximal order; skew field of quotients
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\textit{A. W. Chatters}, Math. Proc. Camb. Philos. Soc. 95, 49--54 (1984; Zbl 0541.16001)

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### References:

[1] | DOI: 10.1016/0021-8693(72)90104-4 · Zbl 0239.16003 |

[2] | Moeglin, C. R. Acad. Sci. 282 pp 1269– (1976) |

[3] | Chatters, Rings with Chain Conditions (1980) |

[4] | DOI: 10.1112/jlms/s2-10.3.281 · Zbl 0313.16011 |

[5] | Dixmier, Enveloping Algebras (1977) |

[6] | Maury, Ordres maximaux au sens de K. Asano. 808 (1980) |

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