Some Near-Ring Embeddings

Malone, Joseph J.; Heatherly, Henry E.

doi:10.1093/qmath/20.1.81

Cited by 11 publications

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“…Such a choice is always possible, see for instance [11]. Embed JR in T 0 (G) as in [8]. Then R has a faithful representation on G, an R simple module.…”

Section: Some Applicationsmentioning

confidence: 99%

On the structure of morphism near-rings

Meldrum

1978

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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SynopsisThe connection between the structure of a near-ring and that of the group on which it acts is used to obtain results concerning the structure of near-rings. A generalized R series is defined for an R module, where R is a zero-symmetric left near-ring, and it is shown that all R modules have maximal R series. The idea of a near-ring which annihilates a series is introduced and some easy consequences of the definition are pointed out. Semi-primitive near-rings are introduced and a general structural result connecting the last two ideas is given. Some special cases which generalize earlier results on endomorphism near-rings are stated. Finally some of the limitations of the idea of semi-primitive near-rings are shown, and some applications are given, in particular to the endomorphism near-rings of soluble groups and of the symmetric groups.

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“…Such a choice is always possible, see for instance [11]. Embed JR in T 0 (G) as in [8]. Then R has a faithful representation on G, an R simple module.…”

Section: Some Applicationsmentioning

confidence: 99%

On the structure of morphism near-rings

Meldrum

1978

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…near-ring generated by the semigroup S of endomorphisms of Fr(X, S). J-D-P-Meldrum [6] The following theorem generalizes Theorem 2.1 of [3]. THEOREM …”

Section: Free (R S) Groupsmentioning

confidence: 87%

The representation of d. g. near-rings

Meldrum

1973

J. Aust. Math. Soc.

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In this paper we study the problem of the representation of d.g. near-rings, and in particular the problem of a faithful representation, which is equivalent to the adjoining of an identity. This problem has been considered by Malone [5] and Malone and Heatherly [6] and [7]. They have shown that a finite near-ring with two sided zero can be embedded in the d.g. near-ring generated by the inner automorphisms of a suitable group, and that an identity can always be adjoined to a near-ring with two sided zero. They have also given some special conditions under which a faithful representation of a d.g. near-ring exists.From another point of view, Frohlich has studied groups over a d.g. nearring in [3] and [4J. If {R, S) is a d.g. near-ring, where S is the distributive semigroup generating R, then he showed that free (R, S) groups exist. We use free (R,S) groups to show that not every d.g. near-ring (R, S) can have a faithful representation on a group, if we insist that S should be a semigroup of distributive elements, i.e. endomorphisms on the group. This is true even in the finite case.We start by setting the work of Frohlich on free (R. S) groups in the context of varieties, using methods differing substantially from his. Using these ideas, we construct in each non-abelian variety a d.g. near-ring without a faithful representation. This opens up the problem of determining those d.g. near-rings which do have a faithful representation. It also leaves open the question of whether it is possible to embed a d.g. near-ring {R, S) in a d.g. near-ring with identity, if we do not insist that the elements of S should be distributive in the larger near-ring.We finish by establishing that for every d.g. near-ring (R,S), there exist 'nearest' d.g. near-rings (R,S), (R,S) which have faithful representations and such that (R,S) is a homomorphic image of (R,S) and (R,S) is a homomorphic image of (R, S). For those d.g. near-rings to which an identity can be adjoined, there is a natural way of doing it. If the near-ring is a ring, then it is interesting to note that this method of adjoining the identity is the standard one. 467 468 • J. D. P. Meldrum [2] Definitions and preliminary results[3]The representation of d.g. near rings 469

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“…T 0 (G) serves as a motivational example of a near-ring. Furthermore every near-ring can be embedded in a T 0 (G) for some G [5].…”

Section: Preliminaries On Near-ringsmentioning

confidence: 99%

“…(1) IfGL = G,thenP <=L, (2) IfGL = G and G is finite, then L = T 0 (G), (3) If G is a finite simple group, then T 0 (G) has no proper left ideals, (4) If L is also a right T 0 (G)-subgroup, then P ^ L and if in addition G is finite, then L = T 0 (G), (5) IfG is finite, then T 0 (G) is a simple near-ring, (6) If G is a simple group, then P is contained in every non-zero left ideal of T 0 (G).…”

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confidence: 99%

One-sided ideals in near-rings of transformations

Heatherly

1972

J. Aust. Math. Soc.

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Let (G, +) be an arbitrary group and let T0(G) = {f∈ Map(G, G): 0f = 0}; the system composed of T0(G) and the operations of pointwise addition and composition of functions form a (left) near-ring. Berman and Silverman, in their investigation of near-rings of transformations [3], found that for every group G the associated near-ring of transformations T0(G) has no proper ideals. In the present paper left and right ideals of T0(G) are considered.

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Some Near-Ring Embeddings

Cited by 11 publications

References 0 publications

On the structure of morphism near-rings

On the structure of morphism near-rings

The representation of d. g. near-rings

One-sided ideals in near-rings of transformations

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