Positive Derivations on Archimedean Almost f-Rings |
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Authors: | Boulabiar Karim |
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Affiliation: | (1) Institut Préparatoire aux Etudes Scientifiques et Techniques, Université du 7 Novembre à Carthage, PB 51, 2070 La Marsa, Tunisia |
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Abstract: | It is shown by P. Colville, G. Davis and K. Keimel that if R is an Archimedean f-ring then a positive group endomorphism D on R is a derivation if and only if the range of D is contained in N(R) and the kernel of D contains R
2, where N(R) is the set of all nilpotent elements in R and R
2 is the set of all products uv in R. The main objective of this paper is to establish the result corresponding to the Colville–Davis–Keimel theorem in the almost f-ring case. The result obtained in this regard is that if D is a positive derivation in an Archimedean almost f-ring, then the range of D is contained in N(R) and the kernel of D contains R
3, where R
3 is the set of all products uvw in R. Examples are produced showing that, contrary to the f-ring case, the converse is in general false and the third power is the best possible. |
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Keywords: | almost f-ring Archimedean f-ring positive derivation |
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