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A Note on the Σ(S)-Injectivity of R(S)

Published online by Cambridge University Press:  20 November 2018

John K. Luedeman
John K. Luedeman*
Affiliation:
Clemson University, Clemson, South Carolina
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Let R be a ring with 1. All modules considered are to be unital left R-modules unless otherwise noted.

Definition. A σ-set for R is a nonempty set Σ of left ideals of R satisfying the following conditions:

  1. (σ1) If I ∊ Σ, J is a left ideal of I, and J ⊇ I, then J ∊ Σ.

  2. (σ2) If I ∊ Σ and r ∊ R, then Ir-1 = {s ∊ R | sr ∊ I} ∊ Σ.

  3. (σ3) If I is a left ideal of R, J ∊ Σ, and It-1 ∊ Σ for each t ∊ J, then I ∊ Σ.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 13 , Issue 4 , December 1970 , pp. 481 - 489
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Vol. 1, Math. Surveys no. 7, Amer. Math. Soc., Providence, R. I., 1961.Google Scholar
2. Connell, I. G., On the group ring, Canad. J. Math. 15 (1963), 650-685.Google Scholar
3. Luedeman, John K., A generalization of the concept of a ring of quotients, Canad. Math. Bull, (to appear).Google Scholar
4. Sanderson, D. F., A generalization of divisibility and injectivity in modules, Canad. Math Bull. 8 (1965), 505-513.Google Scholar
5. Utumi, Y., On continuous rings andself-injective rings, Trans. Amer. Math. Soc. 118 (1965), 158-173.Google Scholar
6. Wenger, R., Self-injective semigroup rings for finite inverse semigroups, Proc. Amer. Math. Soc. 20(1969), 213-216.Google Scholar