Abstract
In this article we introduce the notion of a controlled group graded ring. Let G be a group, with identity element e, and let R = ⊕ g∈G R g be a unital G-graded ring. We say that R is G-controlled if there is a one-to-one correspondence between subsets of the group G and (mutually non-isomorphic) R e -sub-bimodules of R, given by G ⊇ H↦ ⊕ h∈H R h . For strongly G-graded rings, the property of being G-controlled is stronger than that of being simple. We provide necessary and sufficient conditions for a general G-graded ring to be G-controlled. We also give a characterization of strongly G-graded rings which are G-controlled. As an application of our main results we give a description of all intermediate subrings T with R e ⊆ T ⊆ R of a G-controlled strongly G-graded ring R. Our results generalize results for artinian skew group rings which were shown by Azumaya 70 years ago. In the special case of skew group rings we obtain an algebraic analogue of a recent result by Cameron and Smith on bimodules in crossed products of von Neumann algebras.
References
Azumaya, G.: New foundation of the theory of simple rings. Proc. Japan Acad. 22(11), 325–332 (1946)
Cameron, J., Smith, R.R.: Bimodules in crossed products of von Neumann algebras. Adv. Math. 274, 539–561 (2015)
Crow, K.: Simple regular skew group rings. J. Algebra Appl. 4(2), 127–137 (2005)
Dade, E.C.: Group-graded rings and modules. Math. Z. 174, 241–262 (1980)
Jespers, E.: Simple graded rings. Comm. Algebra 21(7), 2437–2444 (1993)
Karpilovsky, G.: The Algebraic Structure of Crossed Products, North-Holland Mathematics Studies, 142. Notas De Matemática [Mathematical Notes], vol. 118, p x+348. North-Holland Publishing Co., Amsterdam (1987). ISBN: 0-444-70239-3
Montgomery, S.: Fixed Rings of Finite Automorphism Groups of Associative Rings, Lecture Notes in Mathematics, vol. 818, p vii+126. Springer, Berlin (1980). ISBN: 3-540-10232-9
Nastasescu, C., Van Oystaeyen, F.: Methods of Graded Rings, Lecture Notes in Mathematics, vol. 1836, p xiv+304. Springer, Berlin (2004). ISBN: 3-540-20746-5
Nauwelaerts, E., Van Oystaeyen, F.: Introducing crystalline graded algebras. Algebr. Represent. Theory 11(2), 133–148 (2008)
Nystedt, P., Öinert, J., Pinedo, H.: Epsilon-strongly graded rings, separability and semisimplicity, arXiv:1606.07592 [math.RA]
Öinert, J.: Simple Group Graded Rings and Maximal Commutativity, Operator Structures and Dynamical Systems (Leiden, NL, 2008), 159–175, Contemp. Math, vol. 503. Amer. Math. Soc., Providence, RI (2009)
Öinert, J.: Simplicity of skew group rings of abelian groups. Comm. Algebra 42 (2), 831–841 (2014)
Öinert, J., Lundström, P.: The ideal intersection property for groupoid graded rings. Comm. Algebra 40(5), 1860–1871 (2012)
Schmid, P.: Clifford theory of simple modules. J. Algebra 119(1), 185–212 (1988)
Van Oystaeyen, F.: On Clifford systems and generalized crossed products. J. Algebra 87(2), 396–415 (1984)
Acknowledgements
The author would like to thank an anonymous referee for valuable comments on the manuscript and for kindly having provided Example 1.
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Presented by Kenneth Goodearl.
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Öinert, J. Bimodules in Group Graded Rings. Algebr Represent Theor 20, 1483–1494 (2017). https://doi.org/10.1007/s10468-017-9696-x
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DOI: https://doi.org/10.1007/s10468-017-9696-x
Keywords
- Graded ring
- Strongly graded ring
- Crossed product
- Skew group ring
- Bimodule
- Picard group
Mathematics Subject Classification (2010)
- 16S35
- 16W50
- 16D40