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Local Near-Rings Of Cardinality P2

Published online by Cambridge University Press:  20 November 2018

Carlton J. Maxson
Carlton J. Maxson*
Affiliation:
State University College, Fredonia, New York 14063
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The main result of this paper is the determination of all nonisomorphic local near-rings <N, +, •> with <N, +> =<C(p) × C(p), +> which are not near-fields. Together with the fundamental paper [6] by Zassenhaus on near-fields and the corollary to Theorem 1 of [2], this 2 paper gives a complete description of all local near-rings of order p2.

We recall that a unitary near-ring N is called local if the subset L of elements in N without left inverses is an (N, N)-subgroup and N ≠ J(N). (J(N) denotes the radical of N given in [1].) In [3] it was proved that N ≠ J(N) whenever L is an ideal of N. (For previous result s concerning local near-rings we refer the reader to [3].)

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 11 , Issue 4 , October 1968 , pp. 555 - 561
Copyright
Copyright © Canadian Mathematical Society 1968

References

1. Beidleman, James C., A radical for near-ring modules. Mich. Math. J., 12(1965) 377-383.Google Scholar
2. Clay, James R. and Malone, Joseph J. Jr The near-rings with identities on certain finite groups. Math. Scand. 19 (1966) 146-150.Google Scholar
3. Maxson, Carlton J., On local near-rings. Math. Z., 106 (1968) 197-205.Google Scholar
4. Maxson, Carlton J., On the construction of finite local near-rings (I): On non-cyclic abelian p-groups (to appear).Google Scholar
5. Zassenhaus, Hans J., The theory of groups, 2nd ed. (Chelsea, New York, 1958).Google Scholar
6. Zassenhaus, Hans J., Über endliche Fastk örper. Abh. Math. Sem., Univ. Hamburg, Vol. II (1936) 187-220.Google Scholar