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Semiendomorphisms of simple near-rings


Authors: Kirby C. Smith and Leon van Wyk
Journal: Proc. Amer. Math. Soc. 115 (1992), 613-627
MSC: Primary 16Y30
DOI: https://doi.org/10.1090/S0002-9939-1992-1081701-7
MathSciNet review: 1081701
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Abstract: Let $ N$ be a finite simple centralizer near-ring that is not an exceptional near-field. A semiendomorphism of $ N$ is a map ' from $ N$ into $ N$ such that $ (a + b)' = a' + b',(aba)' = a'b'a'$, and $ 1' = 1$ for all $ a,b \in N$. It is shown that every semiendomorphism of $ N$ is an automorphism of $ N$. A Jordan-endomorphism of $ N$ is a map ' from $ N$ into $ N$ such that $ (a + b)' = a' + b',(ab + ba)' = a'b' + b'a'$, and $ 1' = 1$ for all $ a,b \in N$. It is shown that every Jordan-endomorphism of $ N$ is an automorphism assuming $ 2 \in N$ is invertible. The above results imply that every semiendomorphism (Jordan-endomorphism) of a "special" class of semisimple near-rings is an automorphism. These results are in contrast to the ring situation where semiendomorphisms tend to be either an automorphism or an antiautomorphism.


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DOI: https://doi.org/10.1090/S0002-9939-1992-1081701-7
Article copyright: © Copyright 1992 American Mathematical Society

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