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A note on semi-homomorphisms of rings

Published online by Cambridge University Press:  17 April 2009

Y. Fong  and
L. van Wyk
Y. Fong
Affiliation:
Department of MathematicsNational Cheng Kung University70102 Tainan, Taiwan, Republic of China
L. van Wyk
Affiliation:
Department of MathematicsUniversity of Stellenbosch7600 StellenboschRepublic of South Africa
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Abstract

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Huq presented a general study of semi-homomorphisms of rings, following, amongst others, Kaplansky's study of semi-automorphisnis of rings and Herstein's study of semi-homomorphisms of groups. Huq gave several “sufficient” conditions for a semi-homomorphism and a semi-monomorphism of rings to be a homomorphism and a monomorphism respectively. In this note we introduce semi-subgroups of groups, provide counterexamples to four of Huq's assertions and show how a minor, albeit forced, change to one of the conditions of the fourth assertion turns it into a special case of another theorem of Huq's.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 40 , Issue 3 , December 1989 , pp. 481 - 486
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Ancochea, G., ‘Le théorème de von Staudt en geometric projective quaternionienne’, J. Reine Angew. Math. 184 (1942), 192–198.Google Scholar
[2]Herstein, I.N., ‘Semi-homomorphisms of groups’, Canad. J. Math. 20 (1968), 384–388.CrossRefGoogle Scholar
[3]Huq, S.A., ‘Semi-homomorphisms of rings’, Bull. Austral. Math. Soc. 36 (1987), 121–125.CrossRefGoogle Scholar
[4]Kaplansky, I., ‘Semi-homomorphisms of rings’, Duke Math. J. 14 (1947), 521–525.CrossRefGoogle Scholar