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Completely reducible near-rings

Published online by Cambridge University Press:  20 January 2009

A. Oswald
A. Oswald
Affiliation:
Department of Mathematics, Teesside Polytechnic, Middlesbrough, Cleveland, TS1 3BA
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To establish our notation N will always denote a (left) near-ring without any type of multiplicative identity (unless the contrary is stated) satisfying On = 0 for each n ∈ N where 0 is the additive identity of N. A group M, written additively, which admits N as a set of right multipliers is a (right) N-module if a ∈ M, n1, n2 ∈ N implies a(n1 + n2) = an1 + an2 and a(n1n2) = (an1)n2. When N has a two-sided identity, 1, we suppose that a 1 = a for each a ∈ M. A subgroup X of M is an N-subgroup of M if it is an N-module; X is a submodule of M if it is a normal subgroup of M and a ∈ M, x ∈ X. n ∈ N implies (a + x)n – an ∈ X. We denote by SL(M) the set of N-subgroups and by L(M) the set of submodules of M. Since N may be regarded as an N-module we can talk about N-subgroups and submodules of N although we usually call the submodules of N right ideals of N. Other definitions can be found in (6).

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 20 , Issue 3 , March 1977 , pp. 187 - 197
Copyright
Copyright © Edinburgh Mathematical Society 1977

References

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