Nil sets defined through additive maps in prime and semiprime rings

Abstract Department of Mathematical and Computer Sciences, Physical Sciences and Earth Sciences Doctor of Philosophy NIL SETS DEFINED THROUGH ADDITIVE MAPS IN PRIME AND SEMIPRIME RINGS by FRANCESCO AMMENDOLIA In this dissertation, we will focus on analyzing the structure of prime and semiprime rings equipped with additive maps that have nilpotent evaluations on appropriate subsets of the ring in which they are defined. In particular the involved maps on R will be derivations, generalized derivations and generalized skew derivations. We remark that the results in literature show that there is a strong relationship between the behaviour of derivations, generalized derivations and generalized skew derivations in prime and semiprime rings and the structure of the rings. Research on how nilpotent maps behave in these contexts has led to important results, such as the characterization of certain classes of rings and the understanding of their structural properties. In particular there is a close connection between the existence of nilpotent maps in a ring and the potential commutativity of the ring itself. Here is why the idea of studying the potential existence of nilpotent subsets of a prime ring naturally arises. Without any doubt, the two milestones of this line of research reside in the articles by S. Montgomery (1974) and I.N. Herstein (1979), in which, respectively, nilpotent traces and nilpotent derivations in prime rings are analyzed. Many researchers have followed up on the work of Montgomery and Herstein over the years, from various points of view on operator algebras, leading to their generalizations: the concept of generalized derivations and generalized skew derivations having nilpotent values. In all that we will examine and study in the continuation of the thesis, R will be a prime ring of characteristic different from 2, Q_r its right Martindale quotient ring, C its extended centroid. We will characterize the structure of R and any possible form of some additive maps, which are denfined of R, in the case appropriate subsets of R have nilpotent values. More precisely, we firstly study nil sets defined through commuting iterates of generalized derivations F and G of R. We prove that, if L is a non-central Lie ideal of R and n ≥ 1 is a fixed integer such that {F^2(x)x − xG^2(x)}^n = 0 for all x ∈ L, then either R ⊆ M2(K), the ring 2 × 2 matrices over a field K, or one of the following holds: (1) F(x) = xa and G(x) = xc for all x ∈ R with a^2 = c^2 ∈ C; (2) F(x) = xa and G(x) = cx for all x ∈ R with a^2 = c^2; (3) F(x) = ax and G(x) = xc for all x ∈ R with a^2 = c^2 ∈ C; (4) F(x) = ax and G(x) = cx for all x ∈ R with a^2 = c^2 ∈ C. Moreover we will also analyze nil sets defined through generalized derivations having homomorphism-like behavior. In this case we show that, if F and G are two generalized derivations of R, L is a non-central Lie ideal of R and n ≥ 1 is a fixed integer such that (F(xy) − G(x)G(y))^n= 0 for any x, y ∈ L, then there exists λ ∈ C such that F(x) = λ2x and G(x) = λx, for any x ∈ R. Finally, we will study nil sets defined through generalized skew derivations F and G of R that emulate the behavior of Jordan derivations, and prove that if {F(x^2) − G(x)x − xG(x)}^n= 0 for all x ∈ R, then either R ⊆ M2(K), the ring 2×2 matrices over a field K, or one of the following holds: (1) F(x) = G(x) = [x, p], for all x ∈ R. (2) ∃η ∈ C, η ̸= 0 such that F(x) = [x, p] + 2ηx and G(x) = [x, p] + ηx, for all x ∈ R.