Herstein's theory is the study of the nonassociative structures and objects arising from associative rings
Herstein's theory is the study of the nonassociative structures and objects arising from associative rings $R$. Examples are: a) the determination of the structure of the Lie ring $R^-$ (arising from the bracket product $[x,y]:=xy-yx$) and of the Jordan ring $R^+$ (arising from the circle product $x\circ y:=xy+yx$) given the structure of $R$, b) the determination of the Jordan homomorphisms of $R$ (as in Hua's theorem for division rings), c) the determination of the structure of $R$ (e.g., $R$ is commmutative) given the existence of some special kind of derivation (as in Posner's theorems on derivations of prime rings). It is usual to ask for $R$ to have some nice structural properties making identities easy to control or simplify (e.g. $R$ prime or semiprime, $R$ having idempotents with special properties); it is also usual to consider also the case of rings with involution $*$, in which case the respective Lie and Jordan rings of skew and symmetric elements are also studied. Besides the pure handling of identities, other techniques of Herstein's theory are based on the multiplication algebra, the central closure, and the Martindale's rings of quotients. Renowned mathematicians that have worked in this area are Jacobson, Herstein, Posner, Amitsur, Lanski, Benkart, Montgomery, Martindale, Beidar, Bresar, Shestakov, and Zelmanov (among many others).