Simple near-ring centralizers of finite rings

Maxson, C. J.; Smith, Kirby C.

doi:10.1090/s0002-9939-1979-0529202-x

Cited by 12 publications

(27 citation statements)

References 3 publications

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“…when Γ = Ssf is a group of automorphisms of a finite group G ( [5]), or when Γ is a finite ring with 1 and G is a faithful, unital /"-module ( [6]). From a structure theorem due to Betsch [1] we have that a finite near-ring N, which is not a ring, is simple if and only ifN= C(J^; G) where jy is a fixed point free group of automorphisms of a finite group G. (A group Jϊf of automorphisms is fixed point free if the identity map in J>/ is the only element of J$?…”

mentioning

confidence: 99%

On semisimple rings that are centralizer near-rings

Maxson¹,

Pettet²,

Smith³

1982

Pacific J. Math.

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mentioning

confidence: 99%

On semisimple rings that are centralizer near-rings

Maxson¹,

Pettet²,

Smith³

1982

Pacific J. Math.

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“…Then f(g l +g 2 ) = fgi + fg 2 if and only if f(gi+g 2 )(x) = (fSi + fg2)( x ) f°r a U xe G. Hence the result.…”

Section: Em^g)mentioning

confidence: 77%

Distributive elements in centralizer near-rings

Maxson

Meldrum

1987

Proceedings of the Edinburgh Mathematical Society

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Let <G,+> be a group with identity 0 and let S be a semigroup of endomorphisms of G. The set Ms(G)={f:G→G; f(0)=0, fσ=σf, for all σ∈S} with the operations of unction addition and composition is a zero-symmetric near-ring with identity called the centralizer near-ring determined by the pair (S, G). Centralizer near-rings have been studied for many classes of semigroups of endomorphisms. (See [8] and the references given there.) In this paper we continue these investigations into the structure of centralizer near-rings via our study of the relationship between distributive elements in Ms(G) and endomorphisms in Ms(G). More specifically, let N = Ms(G) and let Nd={f∈N; f(g1+g2)=fg1+fg2}, the set of distributive elements in N. Under the operation of function composition, Nd is a semigroup containing the identity map, id. Moreover, Nd contains as a submonoid = {α ∈ End G; ασ=σα for all σ∈S}. Here we determine for certain semigroups S, whether or not = Nd.

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“…This near-ring, sometimes called the centralizer near-ring determined by R and M [11][12][13][14][15], obviously contains the ring E R (M) of all /?-endomorphisms of M: 174 Jutta Hausen and Johnny A. Johnson [2] that is…”

Section: Introductionmentioning

confidence: 99%

Centralizer near-rings that are rings

Hausen¹,

Johnson²

1995

J Aust Math Soc A

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Given an /{-module M, the centralizer near-ring J(R(M~) is the set of all functions / : M -> M with f(xr) = f(x)r for all x e M and r € R endowed with point-wise addition and composition of functions as multiplication. In general, MR(M) is not a ring but is a near-ring containing the endomorphism ring E R (M) of M. Necessary and/or sufficient conditions are derived for MR(M) to be a ring. For the case that R is a Dedekind domain, the /{-modules M are characterized for which (i) MR(M~) is a ring; and (ii) M R (M) = ER(M).It is shown that over Dedekind domains with finite prime spectrum properties (i) and (ii) are equivalent.

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Simple near-ring centralizers of finite rings

Abstract: For a finite ring R with identity and a finite unital Ä-module V we call C(R) = {/: K-> V\f(av) = af(v) for all a 6 R, v e V) the nearring centralizer of R. We investigate the structure of C(R) and obtain a characterization of those rings R for which C(R) is a simple nonring.

Cited by 12 publications

References 3 publications

On semisimple rings that are centralizer near-rings

On semisimple rings that are centralizer near-rings

Distributive elements in centralizer near-rings

Centralizer near-rings that are rings

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