Proceedings of the American Mathematical Society

Author(s): Jeremy Haefner
Journal: Proc. Amer. Math. Soc. 124 (1996), 1013-1021.
MSC (1991): Primary 16D90, 16S35, 16S40, 16S50, 16W50
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Abstract: Let $R$

be a ring graded by a group $G$

. We are concerned with describing those $G$

-graded rings that are graded equivalent to $G$

-crossed products. We give necessary and sufficient conditions for when a strongly graded ring is graded equivalent to a crossed product, provided that the 1-component is either Azumaya or semiperfect. Our result uses the torsion product theorem of Bass and Guralnick. We also construct various examples of such rings.

1.: D. F. Anderson, The kernel of $\Pic(R_0)\to \Pic(R)$ for a graded domain, Mathematical Reports of the Academy of Science, vol. 13, The Royal Society of Canada, 1991, pp. 248--253. MR 92j:13001
2.: F. Anderson and K. Fuller, Rings and categories of modules, Springer-Verlag, New York, 1974. MR 54:5281
3.: H. Bass, Algebraic -theory, Benjamin, New York, 1968. MR 40:2736
4.: H. Bass and R. Guralnick, Projective modules with free multiples and powers, Proc. Amer. Math. Soc. 96 (1986), 207--208. MR 87a:13011
5.: ------, Torsion in the Picard group and extension of scalars, J. Pure Appl. Algebra, 52 (1988), 213--217. MR 89e:13006
6.: M. Beattie, A generalization of the smash product of a graded ring, J. Pure Appl. Algebra 52 (1988), 219--226. MR 89k:16005
7.: M. Cohen and S. Montgomery, Group-graded rings, smash products, and group actions, Trans. Amer. Math. Soc. 282 (1984), 237--258. MR 85i:16002
8.: C. Curtis and I. Reiner, Methods of representation theory with applications to finite groups and orders, Vol. 2, Wiley, New York, 1987. MR 88f:20002
9.: E. Dade, Group-graded rings and modules, Math. Z. 174 (1980), 241--262. MR 82c:16028
10.: R. Guralnick, Bimodules over pi rings, Methods in Module Theory, Marcel Dekker, New York, 1993, pp. 117--134. MR 94a:16001
11.: R. Guralnick and S. Montgomery, On invertible bimodules and automorphisms of noncommutative rings, Trans. Amer. Math. Soc. 341 (1994), 917--937. MR 94d:16027
12.: R. Gordon and E. Green, Graded Artin algebras, J. Algebra 76 (1982), 111--137. MR 83m:16028a
13.: J. Haefner, Graded Morita theory for infinite groups, J. Algebra 169 (1994), 552--586. MR 95j:16051
14.: ------, Graded equivalence theory with applications, J. Algebra 172 (1995), 385--423. CMP 95:09
15.: ------, A strongly graded ring that is not graded equivalent to a skew group ring, Comm. Algebra 22 (1994), 4795--4799. MR 95c:16055
16.: C. Menini and C. N\u{a}st\u{a}sescu, When is R-gr equivalent to the category of modules? J. Pure Appl. Algebra 51 (1988), 277--291. MR 89f:16002
17.: D. Passman, Infinite crossed products, Academic Press, New York, 1989. MR 90g:16002
18.: D. Quinn, Group-graded rings and duality, Trans. Amer. Math. Soc. 292 (1985), 155--167. MR 87d:16002
19.: A. del Río, Graded rings and equivalences of categories, Comm. Algebra 19 (1991), 997--1012. MR 92d:16011

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