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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Boulabiar, K. Positive Derivations on Archimedean Almost f-Rings. Order 19, 385–395 (2002). https://doi.org/10.1023/A:1022869819129
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DOI: https://doi.org/10.1023/A:1022869819129