caa a22 4500 386372039 CHVBK 20180307112001.0 cr unu---uuuuu 161130e198901 xx s 000 0 eng 10.1017/S001708950000759X doi S001708950000759X pii (NATIONALLICENCE)cambridge-10.1017/S001708950000759X Jordan D. A. Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH Unique factorisation of normal elements in non-commutative rings [Elektronische Daten] [D. A. Jordan] In the literature there are several generalisations to non-commutative rings of the notion of a unique factorisation domain from commutative algebra. This paper follows in the spirit of [1, 3] and is set in the context of Noetherian rings. In [3], A. W. Chatters and the author denned a Noetherian UFR (unique factorisation ring) to be a prime Noetherian ring R in which every non-zero prime ideal contains a prime ideal generated by a non-zero normal element p, that is, by an element p such that pR = Rp. The class of Noetherian UFRs includes the Noetherian UFDs studied by Chatters in [1], while a commutative Noetherian ring is a UFR if and only if it is a UFD in the usual sense. For a Noetherian UFR R, the following are simple consequences of the definition: (i) every non-zero ideal of R contains a non-zero normal element; (ii) the set N(R) of non-zero normal elements of R is a unique factorisation monoid in the sense of [4, Chapter 3]. Copyright © Glasgow Mathematical Journal Trust 1989 Glasgow Mathematical Journal Cambridge University Press 31/1(1989-01), 103-113 0017-0895 31:1<103 1989 31 GMJ https://doi.org/10.1017/S001708950000759X text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S001708950000759X text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Jordan D. A. Department of Pure Mathematics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH NATIONALLICENCE 773 0- Glasgow Mathematical Journal Cambridge University Press 31/1(1989-01), 103-113 0017-0895 31:1<103 1989 31 GMJ CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence SWISSBIB 386216975 BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge