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


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.)
Full Text: DOI


[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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