A space of pseudoquotients denoted by B(X, S) is defined as equivalence classes of pairs (x, f); where x is an element of a nonempty set X, f is an element of S; a commutative semigroup of injective maps from X to X; and (x, f) ~ (y, g) for gx = fy: If X is a ring and elements of S are ring homomorphisms, then B(X, S) is a ring. We show that, under natural conditions, a derivation on X has a unique extension to a derivation on B(X, S): We also consider (α, β) -Jordan derivations, inner derivations, and generalized derivations.
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References
D. Atanasiu, P. Mikusiński, and D. Nemzer, “An algebraic approach to tempered distributions,” J. Math. Anal. Appl., 384, 307–319 (2011).
Li Jiankui and Jiren Zhou, “Characterization of Jordan derivations and Jordan homomorphisms,” Linear and Multilinear Algebra, 52, No. 2, 193–204 (2011).
J. A. Erdos, “Operator of finite rank in nest algebras,” London Math. Soc., 43, 391–397 (1968).
L. B. Hadwin, “Local multiplications on algebras spanned by idempotents,” Lin. Multilin. Algebras, 37, 259–263 (1994).
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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 65, No. 6, pp. 863–869, June, 2013.
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Majeed, A., Mikusiński, P. Derivations on Pseudoquotients. Ukr Math J 65, 959–966 (2013). https://doi.org/10.1007/s11253-013-0833-3
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DOI: https://doi.org/10.1007/s11253-013-0833-3