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Factorizations in Bounded Hereditary Noetherian Prime Rings

Part of: Chain conditions, growth conditions, and other forms of finiteness Semigroups Conditions on elements Homological methods Grothendieck groups and $K_0$

Published online by Cambridge University Press:  19 November 2018

Daniel Smertnig
Daniel Smertnig*
Affiliation:
University of Graz, NAWI Graz, Institute for Mathematics and Scientific Computing, Heinrichstraße 36, 8010 Graz, Austria ([email protected])
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Abstract

If H is a monoid and a = u1 ··· uk ∈ H with atoms (irreducible elements) u1, … , uk, then k is a length of a, the set of lengths of a is denoted by Ⅼ(a), and ℒ(H) = {Ⅼ(a) | a ∈ H} is the system of sets of lengths of H. Let R be a hereditary Noetherian prime (HNP) ring. Then every element of the monoid of non-zero-divisors R• can be written as a product of atoms. We show that if R is bounded and every stably free right R-ideal is free, then there exists a transfer homomorphism from R• to the monoid B of zero-sum sequences over a subset Gmax(R) of the ideal class group G(R). This implies that the systems of sets of lengths, together with further arithmetical invariants, of the monoids R• and B coincide. It is well known that commutative Dedekind domains allow transfer homomorphisms to monoids of zero-sum sequences, and the arithmetic of the latter has been the object of much research. Our approach is based on the structure theory of finitely generated projective modules over HNP rings, as established in the recent monograph by Levy and Robson. We complement our results by giving an example of a non-bounded HNP ring in which every stably free right R-ideal is free but which does not allow a transfer homomorphism to a monoid of zero-sum sequences over any subset of its ideal class group.

Keywords

Hereditary Noetherian prime ringsDedekind prime ringsKrull monoidsfactorization theorymonoids of zero-sum sequencessets of lengthscatenary degreestransfer homomorphismsideal class groups

MSC classification

Primary: 16E60: Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc.
Secondary: 16P40: Noetherian rings and modules 16U30: Divisibility, noncommutative UFDs 19A49: $K_0$ of other rings 20M13: Arithmetic theory of monoids
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 62 , Issue 2 , May 2019 , pp. 395 - 442
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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References

1.Akalan, E. and Marubayashi, H., Multiplicative ideal theory in non-commutative rings, In Multiplicative ideal theory and factorization theory (eds Chapman, S. T., Fontana, M., Geroldinger, A. and Olberding, B.), pp. 1–21 (Springer International Publishing, 2016).Google Scholar
2.Anderson, D. D. (ed.), Factorization in integral domains, Lecture Notes in Pure and Applied Mathematics, Volume 189 (Marcel Dekker Inc., New York, 1997).Google Scholar
3.Bachman, D., Baeth, N. R. and Gossell, J., Factorizations of upper triangular matrices, Linear Algebra Appl. 450 (2014), 138–157.Google Scholar
4.Baeth, N. R. and Smertnig, D., Factorization theory: from commutative to noncommutative settings, J. Algebra 441 (2015), 475–551.Google Scholar
5.Baeth, N. R. and Wiegand, R., Factorization theory and decompositions of modules, Amer. Math. Mon. 120(1) (2013), 3–34.Google Scholar
6.Baeth, N. R., Ponomarenko, V., Adams, D., Ardila, R., Hannasch, D., Kosh, A., McCarthy, H. and Rosenbaum, R., Number theory of matrix semigroups, Linear Algebra Appl. 434(3) (2011), 694–711.Google Scholar
7.Bell, J. P., Heinle, A. and Levandovskyy, V., On noncommutative finite factorization domains, Trans. Amer. Math. Soc. 369(4) (2017), 2675–2695.Google Scholar
8.Chapman, S. T. (ed.), Arithmetical properties of commutative rings and monoids, Lecture Notes in Pure and Applied Mathematics, Volume 241 (Chapman & Hall/CRC, Boca Raton, FL, 2005).Google Scholar
9.Chapman, S. T., Fontana, M., Geroldinger, A. and Olberding, B. (ed.), Multiplicative ideal theory and factorization theory, Commutative and Non-commutative Perspectives (Springer International Publishing, 2016).Google Scholar
10.Estes, D. R., Factorization in hereditary orders, Linear Algebra Appl. 157 (1991), 161–164.Google Scholar
11.Estes, D. R. and Matijevic, J. R., Matrix factorizations, exterior powers, and complete intersections, J. Algebra 58(1) (1979), 117–135.Google Scholar
12.Estes, D. R. and Matijevic, J. R., Unique factorization of matrices and Towber rings, J. Algebra 59(2) (1979), 387–394.Google Scholar
13.Facchini, A., Direct-sum decompositions of modules with semilocal endomorphism rings, Bull. Math. Sci. 2(2) (2012), 225–279.Google Scholar
14.Facchini, A., Smertnig, D. and Khanh Tung, N., Cyclically presented modules, projective covers and factorizations, In Ring theory and its applications, Contemporary Mathematics, Volume 609, pp. 89–106 (American Mathematical Society, Providence, RI, 2014).Google Scholar
15.Fontana, M., Houston, E. and Lucas, T., Factoring ideals in integral domains, Lecture Notes of the Unione Matematica Italiana, Volume 14 (Springer, Heidelberg, 2013).Google Scholar
16.Geroldinger, A., Additive group theory and non-unique factorizations, In Combinatorial number theory and additive group theory, Advanced Courses in Mathematics, CRM Barcelona, pp. 1–86 (Birkhäuser Verlag, Basel, 2009).Google Scholar
17.Geroldinger, A., Non-commutative Krull monoids: a divisor theoretic approach and their arithmetic, Osaka J. Math. 50(2) (2013), 503–539.Google Scholar
18.Geroldinger, A., Sets of lengths, Amer. Math. Mon. 123(10) (2016), 960–988.Google Scholar
19.Geroldinger, A. and Halter-Koch, F., Non-unique factorizations: algebraic, combinatorial and analytic theory, Pure and Applied Mathematics, Volume 278 (Chapman and Hall/CRC, Boca Raton, FL, 2006).Google Scholar
20.Grynkiewicz, D. J., Structural additive theory, Developments in Mathematics, Volume 30 (Springer, Cham, 2013).Google Scholar
21.Goodearl, K. R. and Yakimov, M. T., Quantum cluster algebras and quantum nilpotent algebras, Proc. Natl. Acad. Sci. USA 111(27) (2014), 9696–9703.Google Scholar
22.Hallouin, E. and Maire, C., Cancellation in totally definite quaternion algebras, J. Reine Angew. Math. 595 (2006), 189–213.Google Scholar
23.Lam, T. Y., Serre's problem on projective modules, Springer Monographs in Mathematics (Springer-Verlag, Berlin, 2006).Google Scholar
24.Launois, S., Lenagan, T. H. and Rigal, L., Quantum unique factorisation domains, J. Lond. Math. Soc. (2) 74(2) (2006), 321–340.Google Scholar
25.Leuschke, G. J. and Wiegand, R., Cohen-Macaulay representations, Mathematical Surveys and Monographs, Volume 181 (American Mathematical Society, Providence, RI, 2012).Google Scholar
26.Levy, L. S. and Robson, J. C., Hereditary Noetherian prime rings and idealizers, in Mathematical Surveys and Monographs, Volume 174 (American Mathematical Society, Providence, RI, 2011).Google Scholar
27.McConnell, J. C. and Robson, J. C., Homomorphisms and extensions of modules over certain differential polynomial rings, J. Algebra 26 (1973), 319–342.Google Scholar
28.McConnell, J. C. and Robson, J. C., Noncommutative Noetherian rings, Graduate Studies in Mathematics, Volume 30, revised edn (American Mathematical Society, Providence, RI, 2001).Google Scholar
29.Narkiewicz, W., Elementary and analytic theory of algebraic numbers, Springer Monographs in Mathematics, 3rd edn (Springer-Verlag, Berlin, 2004).Google Scholar
30.Reiner, I., Maximal orders, London Mathematical Society Monographs, Volume 5 (Academic Press, London–New York, 1975).Google Scholar
31.Rump, W., Invertible ideals and noncommutative arithmetics, Commun. Algebra 29(12) (2001), 5673–5686.Google Scholar
32.Rump, W. and Yang, Y., Hereditary arithmetics, J. Algebra 468 (2016), 214–252.Google Scholar
33.Smertnig, D., Sets of lengths in maximal orders in central simple algebras, J. Algebra 390 (2013), 1–43.Google Scholar
34.Smertnig, D., A note on cancellation in totally definite quaternion algebras, J. Reine Angew. Math. 707 (2015), 209–216.Google Scholar
35.Smertnig, D., Factorizations of elements in noncommutative rings: a survey, Multiplicative Ideal Theory and Factorization Theory, pp. 353–402 (Springer International Publishing, 2016).Google Scholar
36.Vignéras, M.-F., Simplification pour les ordres des corps de quaternions totalement définis, J. Reine Angew. Math. 286/287 (1976), 257–277.Google Scholar