Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Scalar-reflexive rings

Authors: Don Hadwin and Jeanne Wald Kerr
Journal: Proc. Amer. Math. Soc. 103 (1988), 1-8
MSC: Primary 13C13; Secondary 13E10
MathSciNet review: 938634
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A module over a commutative ring is scalar-reflexive if the only endormorphisms that leave invariant every submodule are scalars. We investigate the class of rings for which every module is scalar-reflexive and the class of rings for which every finitely generated module is scalar-reflexive. For a certain class of rings we show that these properties are equivalent to every finitely generated module being a direct sum of cyclic modules.

References [Enhancements On Off] (What's this?)

  • [1] E. Azoff, $ K$-reflexivity in finite-dimensional spaces, Duke Math. J. 40 (1973), 821-830. MR 0331081 (48:9415)
  • [2] L. Brickman and P. A. Fillmore, The invariant subspace lattice of a linear transformation, Canad. J. Math. 19 (1967), 810-822. MR 0213378 (35:4242)
  • [3] D. W. Hadwin, Algebraically reflexive linear transformations, Linear and Multilinear Alg. 14 (1983), 139-147. MR 718951 (85e:47003)
  • [4] I. N. Herstein, Noncommutative rings, Carus Math. Monographs, No. 15, Math. Assoc. of Amer., 1973. MR 1449137 (97m:16001)
  • [5] N. Jacobson, Basic algebra. II, Freeman, San Francisco, Calif., 1980. MR 571884 (81g:00001)
  • [6] I. Kaplansky, Infinite abelian groups, Univ. of Michigan Press, Ann Arbor, 1954. MR 0065561 (16:444g)
  • [7] -, Fields and rings, Univ. of Chicago Press, Chicago, 1972. MR 0349646 (50:2139)
  • [8] R. Wiegand and S. Wiegand, Commutative rings whose finitely generated modules are direct sums of cyclics, Abelian Group Theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M.), Lecture Notes in Math., vol. 616, Springer-Verlag, Berlin, 1977, pp. 406-423. MR 0498528 (58:16635)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13C13, 13E10

Retrieve articles in all journals with MSC: 13C13, 13E10

Additional Information

Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society