On reflexivity of direct sums

Camillo, V.; Fuller, K.

doi:10.1090/S0002-9939-00-05331-4

On reflexivity of direct sums
HTML articles powered by AMS MathViewer

by V. P. Camillo and K. R. Fuller PDF

Proc. Amer. Math. Soc. 128 (2000), 2855-2862 Request permission

Abstract:

Necessary and sufficient conditions are presented to insure that the direct sum of two reflexive representations of a finite dimensional algebra is reflexive, and it is shown that for each such algebra, there is an integer $k$ such that the direct sum of $k$ copies of each of its representations is reflexive. Given a ring $\Delta ,$ our results are actually presented in the more general setting of $\Delta$-representations of a ring $R.$

References

Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, 2nd ed., Graduate Texts in Mathematics, vol. 13, Springer-Verlag, New York, 1992. MR 1245487, DOI 10.1007/978-1-4612-4418-9
Edward A. Azoff, $K$-reflexivity in finite dimensional spaces, Duke Math. J. 40 (1973), 821–830. MR 331081
Edward A. Azoff and Marek Ptak, On rank two linear transformations and reflexivity, J. London Math. Soc. (2) 53 (1996), no. 2, 383–396. MR 1373068, DOI 10.1112/jlms/53.2.383
Sheila Brenner and M. C. R. Butler, Endomorphism rings of vector spaces and torsion free abelian groups, J. London Math. Soc. 40 (1965), 183–187. MR 174593, DOI 10.1112/jlms/s1-40.1.183
James A. Deddens, Every isometry is reflexive, Proc. Amer. Math. Soc. 28 (1971), 509–512. MR 278099, DOI 10.1090/S0002-9939-1971-0278099-7
J. A. Deddens and P. A. Fillmore, Reflexive linear transformations, Linear Algebra Appl. 10 (1975), 89–93. MR 358390, DOI 10.1016/0024-3795(75)90099-3
Lifeng Ding, Separating vectors and reflexivity, Linear Algebra Appl. 174 (1992), 37–52. MR 1176449, DOI 10.1016/0024-3795(92)90040-H
Kent R. Fuller, Algebras from diagrams, J. Pure Appl. Algebra 48 (1987), no. 1-2, 23–37. MR 915987, DOI 10.1016/0022-4049(87)90105-8
K. R. Fuller, W. K. Nicholson, and J. F. Watters, Reflexive bimodules, Canad. J. Math. 41 (1989), no. 4, 592–611. MR 1012618, DOI 10.4153/CJM-1989-026-x
K. R. Fuller, W. K. Nicholson, and J. F. Watters, Direct sums of reflexive modules, Linear Algebra Appl. 239 (1996), 201–214. MR 1384923, DOI 10.1016/S0024-3795(96)90012-9
K. R. Fuller, W. K. Nicholson, and J. F. Watters, On $k$-reflexive representations of algebras, Proc. Amer. Math. Soc. 125 (1997), no. 1, 47–50. MR 1363461, DOI 10.1090/S0002-9939-97-03666-6
Javad Faghih Habibi and William H. Gustafson, Reflexive serial algebras, Linear Algebra Appl. 99 (1988), 217–223. MR 925158, DOI 10.1016/0024-3795(88)90133-4
Don Hadwin, Algebraically reflexive linear transformations, Linear and Multilinear Algebra 14 (1983), no. 3, 225–233. MR 718951, DOI 10.1080/03081088308817559
P. R. Halmos, Reflexive lattices of subspaces, J. London Math. Soc. (2) 4 (1971), 257–263. MR 288612, DOI 10.1112/jlms/s2-4.2.257
N. Snashall, Reflexivity of modules over QF-3 algebras, Comm. in Algebra 26 (1998), 4233–4242.