Factorizations of elements in noncommutative rings: a survey. (English) Zbl 1356.16029
Chapman, Scott (ed.) et al., Multiplicative ideal theory and factorization theory. Commutative and non-commutative perspectives. Selected papers based on the presentations at the meeting ‘Arithmetic and ideal theory of rings and semigroups’, Graz, Austria, September 22–26, 2014. Cham: Springer (ISBN 978-3-319-38853-3/hbk; 978-3-319-38855-7/ebook). Springer Proceedings in Mathematics & Statistics 170, 353-402 (2016).
Summary: We survey results on factorizations of non-zero-divisors into atoms (irreducible elements) in noncommutative rings. The point of view in this survey is motivated by the commutative theory of nonunique factorizations. Topics covered include unique factorization up to order and similarity, 2-firs, and modular LCM domains, as well as UFRs and UFDs in the sense of A. W. Chatters and D. A. Jordan [J. Lond. Math. Soc., II. Ser. 33, 22–32 (1986; Zbl 0601.16001)] and generalizations thereof. We recall arithmetical invariants for the study of nonunique factorizations, and give transfer results for arithmetical invariants in matrix rings, rings of triangular matrices, and classical maximal orders as well as classical hereditary orders in central simple algebras over global fields.
For the entire collection see [Zbl 1346.13002].
For the entire collection see [Zbl 1346.13002].
MSC:
| 16U30 | Divisibility, noncommutative UFDs |
| 16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |
Citations:
Zbl 0601.16001References:
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