A general notion of noncommutative Krull rings. (English) Zbl 0636.16003
Using an idea of Passman, the authors give a characterization of the symmetric maximal quotient ring (and other quotient rings) and they prove that it is indeed left-right symmetric. The notion of a Krull ring as in the sense of Miyashita is reintroduced. This notion is studied in connection with Asano orders, defined within the symmetric Martindale ring of quotients.
Reviewer: C.Năstăsescu
MSC:
| 16U30 | Divisibility, noncommutative UFDs |
| 16P50 | Localization and associative Noetherian rings |
| 16H05 | Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) |
References:
| [1] | Anderson, D. F., Graded Krull domains, Comm. Algebra, 7, 79-106 (1979) · Zbl 0402.13013 |
| [2] | Chamarie, M., Anneaux de Krull non commutatifs, J. Algebra, 72, 210-222 (1981) · Zbl 0468.16008 |
| [3] | Chatters, A. W.; Hajarnavis, C. R., Rings with Chain Conditions (1980), Pitman: Pitman London · Zbl 0446.16001 |
| [4] | Chouinard, L. G., Krull semigroups and divisor class groups, Canad. J. Math., 23, 1459-1468 (1981) · Zbl 0453.13005 |
| [5] | Fossum, R., Maximal orders over Krull domains, J. Algebra, 10, 321-332 (1968) · Zbl 0165.35204 |
| [6] | Jespers, E.; Le Bruyn, L.; Wauters, P., ω-Krull rings I, Comm. Algebra, 10, 17, 1801-1818 (1982) · Zbl 0508.16004 |
| [7] | Kennedy, R., Krull rings, Pacific J. Math., 89, 1, 131-136 (1980) · Zbl 0402.13012 |
| [9] | Marubayashi, H., Polynomial rings over Krull orders in simple Artinian rings, Hokkaido Math. J., 9, 63-78 (1980) · Zbl 0436.16005 |
| [10] | Maury, G.; Raynaud, J., Ordres maximaux au sens de K. Asano, (“Lect. Notes in Math.” No. 808 (1980), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0432.16003 |
| [11] | Miyashita, Y., On a Krull order, Osaka J. Math., 18, 33-54 (1981) · Zbl 0456.16007 |
| [12] | Nǎstǎsescu, C.; Van Oystaeyen, F., Graded and filtered rings and modules, (“Lect. Notes in Math.” No. 758 (1979), Springer-Verlag: Springer-Verlag Berlin) · Zbl 0418.16001 |
| [13] | Nǎstǎsescu, C.; Vand Oystaeyen, F., Graded Ring Theory (1982), North-Holland: North-Holland Amsterdam · Zbl 0494.16001 |
| [14] | Passmman, D. S., The Algebraic Structure of Group Rings (1977), Wiley-Interscience: Wiley-Interscience London · Zbl 0368.16003 |
| [15] | Passman, D. S., Computing the symmetric ring of quotients, J. Algebra, 105, 207-235 (1987) · Zbl 0605.16003 |
| [16] | Rehm, H. P., Multiplicative ideal theory of noncommutative Krull pairs I: module systems, Krull ring-type chain conditions, and application two two-sided ideals, J. Algebra, 48, 150-165 (1977) · Zbl 0372.16003 |
| [17] | Wauters, P., Ω-Krull rings and Gr-Ω-Krull rings in the non-P.I. case, J.Pure Appl. Algebra, 32, 95-113 (1984) · Zbl 0541.16006 |
| [18] | Wauters, P., E. Jespers Asano-orders and graded rings, Comm. Algebra, 13, 4, 811-833 (1985) · Zbl 0559.16001 |
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