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# Presentations of faithful d.g. near-rings

Published online by Cambridge University Press:  20 January 2009

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Many basic definitions and results in the theory of near-rings can be found in G. Pilz (4). We follow these for the most part, except that we use left near-rings rather than right near-rings. We follow exactly an earlier paper, Meldrum (2), where there are detailed definitions and many results relating to faithful d.g. near-rings. Let R be a d.g. near-ring, distributively generated by the semigroup S, which need not be the semigroup of all distributive elements. Denote such a d.g. near-ring by (R, S). Then (R, +) = Gp < S; > where is a set of defining relations in S. Let (T, U) be a d.g. near-ring. Then a d.g. homomorphism θ from (R, S) to (T, U) is a near-ring homomorphism from R to T which satisfies U. If (G, +) is a group, let T0(G) be the near-ring of all maps from G to itself with pointwise addition and map composition. Let End G be the semigroup of all endomorphisms of G. Then (E(G), End G) is a d.g. near-ring. A d.g. near-ring (R, S) is faithful if there exists a d.g. monomorphism θ:(R, S) → (E(G), End G) for some group G.

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## References

#### REFERENCES

(1) Mahmood, S. J., Limits and colimits in categories of d.g. near-rings, Proc. Edinburgh Math. Soc. 23 (1980), 17.Google Scholar
(2) Meldrum, J. D. P., The representation of d.g. near-rings, J. Austral. Math. Soc. 16 (1973), 467480.Google Scholar
(3) Meldrum, J. D. P., On the structure of morphism near-rings, Proc. Royal Soc. Edinburgh 81A (1978), 287298.Google Scholar
(4) Pilz, G., Near-rings (North Holland, Amsterdam, 1976).Google Scholar
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