I have an interest in applications of higher derivations, mainly in prime and semiprime rings.
Have a look at Google https://www.google.gr/search?q=derivations%2C+in+prime+and+semiprime+rings&oq=derivations%2C+in+prime+and+semiprime+rings&aqs=chrome..69i57.13213j0j7&sourceid=chrome&espv=210&es_sm=119&ie=UTF-8#q=derivations,+in+prime+and+semiprime+rings&start=10
On the other hand Antonio Di Nola has some theory of semirings applied to MV-algebras.
see https://www.researchgate.net/profile/Antonio_Nola/contributions/?ev=brs_act
This is a good question.
The study of derivations in prime rings can be traced back to
E.C. Posner, Derivations in prime rings, Proc. Amer. Math. Soc. 8, 1957, 1093-1100:
http://www.ams.org/journals/proc/1957-008-06/S0002-9939-1957-0095863-0/S0002-9939-1957-0095863-0.pdf
A recent paper on derivations in semiprime rings is
B. Dhara, V. De Filippis, Co-commutators with generalized derivations in prime and semiprime rings, Pub. Math. Debrecen 5958, 2014, 1-22:
http://www.math.unideb.hu/publi/forthcoming/5958-Dhara-etal.pdf
Following up on what @Shervin Sahebi observed, see the proof of Theorem 1, page 13.
An important application of prime rings is in the study of noncommtative projective curves. See
J.T. Stafford, M. Van Den Bergh, Noncommutative rings and noncommutative surfaces, Bulletin Amer. Math. Soc. 00, 2000, arXiv: math/9910082v2 [math.RA]:
https://doclib.uhasselt.be/dspace/bitstream/1942/2621/1/9910082v2.pdf
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