A Note on Invariant Basis Number and Types for Strongly Graded Rings
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Abstract
: Given any pair of positive integers (n, k) and any nontrivial finite group G, we show that there exists a ring R of type (n, k) such that R is strongly graded by G and the identity component Re has Invariant Basis Number. Moreover, for another pair of positive integers (n', k') with n ≤ n' and k | k', it is proved that there exists a ring R of type (n, k) such that R is strongly graded by G and Re has type (n', k'). These results were mentioned in [G. Abrams, Invariant basis number and types for strongly graded rings, J. Algebra 237 (2001) 32-37] without proofs.
Keywords:
rongly graded ring, Invariant Basis Number, type.
References
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[2] W.G. Leavitt, The module type of a ring, Trans. Amer. Math. Soc. 103 (1962) 113-130.
[3] C. Năstăsescu, B. Torrecillas, F. Van Oystaeyen, IBN for graded rings, Comm. Algebra 28 (2000) 1351-1360.
[4] G. Abrams, Invariant basis number and types for strongly graded rings, J. Algebra 237 (2001) 32-37.
[5] C. Năstăsescu, F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Mathematics 1836, Springer--Verlag Berlin Heidelberg, 2004.
[6] J.M. Howie, Fundamentals of Semigroup Theory, London Math. Soc. Monographs New Series 12, Oxford University Press, 2003.
[7] G.M. Bergman, Coproducts and some universal ring constructions, Trans. Amer. Math. Soc. 200 (1974) 33-88.
[8] G. Abrams, G. Aranda Pino, The Leavitt path algebra of a graph, J. Algebra 293 (2005) 319-334.
[9] P. Ara, M.A. Moreno, E. Pardo, Nonstable K-theory for graph algebras, Algebr. Represent. Theory 10 (2007) 157-178.
[10] R. Hazrat, The graded structure of Leavitt path algebras, Israel J. Math. 195 (2013) 833-895.