Description
"This outstanding research monograph deals with the nuances of a revolutionary concept of differential identities, which originated with the advent of derivations. The notion of derivation was introduced in rings as an additive map satisfying the Leibnitz rule. In an algebra, the condition of additivity translates to that of linearity. The operations of differentiation on a polynomial ring F[x], with F a field, is a natural example to begin with. In the study of derivations, inner derivations play a pivotal role, that assigns a commutator associated with a single fixed element to each element, i.e., x maps to [a,x]. Inner derivation may sometimes be the only possible derivation on rings or algebras. Among classical algebraic concepts, differential identity seems to be the closest one to the concept of a functional identity and with the advent of the twenty-first century the theory of differential identities has got a tremendous application in solving several outstanding problems in the theory of rings. Coming up with interesting results, the note Differential identities in rings and algebras with applications provides various contexts in which derivations are defined. Though the notion of derivations is quite old, it is still relevant in modern-day math. Posner's breakthrough revolutionized the study of derivations, by relating derivation to that of the structure of rings and algebras. Putting it differently, Posner opened up a door to several new and exciting results, mostly done on prime and semiprime rings. The theory of derivation was taken forward by various stalwarts like Beidar, Bell, Bergen, Bresar, Herstein, Hvala, Lanski, Lee, Luh, Kharchenko, Martindale, Vukman-to mention a few only. They have been keen to emphasize the applicability of derivations in each of their results. No wonder derivation finds its place in the applications of Coding theory, Cryptography, Information science, Field theory, Lie algebras, Linear differential equations, and the list goes on"--
ISBN 9781032824543
