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Injective and projective Boolean-like rings

Part of: General commutative ring theory Homological algebra

Published online by Cambridge University Press:  09 April 2009

V. Swaminathan
V. Swaminathan
Affiliation:
Department of Mathematics Andhra UniversityWaltair - 530 003, India
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Abstract

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A Boolean-like ring R is a commutative ring with unity in which 2x = 0 and xy(1 + x)(1 + y) = 0 hold for all elements x, y of the ring R. It is shown in this paper that in the category of Boolean-like rings, R is injective if and only if R is a complete Boolean ring and R is projective if and only if R = {0, 1}.

Keywords

Category of Boolean-like ringsBoolean-like ringsinjective and projective objectscomplete Boolean rings.

MSC classification

Secondary: 13A99: None of the above, but in this section 18G05: Projectives and injectives
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 33 , Issue 1 , August 1982 , pp. 40 - 49
Copyright
Copyright © Australian Mathematical Society 1982

References

Balbes, R. (1966). ‘Projective and injective distributive lattices’. Notices Amer. Math. Soc. 13. 740.Google Scholar
Balbes, and Horn, A. (1970), ‘Injective and projective Heyting algebras’, Trans. Amer. Math. Soc. 148. 549–560.CrossRefGoogle Scholar
Balbes, R. and Dwinger, P. (1974), Distributive lattices (University of Missouri Press. Columbia, Missouri).Google Scholar
Rao, K. P. S. Baskara and Rao, M. Baskara (1979). ‘The lattice of subalgcbras of a Boolean algebra’. Czechoslovak Math. J. 29. 530–545.CrossRefGoogle Scholar
Cignoli, R. (1975), ‘Injective DeMorgan and Kleene algebras’, Proc. Amer. Math, Soc. 47, 269–278.CrossRefGoogle Scholar
Foster, A. L. (1946), ‘Theory of Boolean-like rings’, Trans. Amer. Math. Soc. 59. 166–187.CrossRefGoogle Scholar
Haines, D. C. (1974), ‘Injective objects in the category of p-rings’, Proc. Amer. Math. Soc. 42. 57–60.Google Scholar
Sikorski, R. (1948), ‘A theorem on extension of homomorphisms’, Ann. Polon. Math. 21, 332–335.Google Scholar
Swaminathan, V. (1980), ‘On Foster's Boolean-like rings’, Math. Seminar Notes, Kohe Unic. 8, 347–367.Google Scholar