Commutativity conditions for rings: 1950–2005

Annihilator Conditions of Multiplicative Reverse Derivations on Prime Rings

“…Substituting y with ry in (19), we obtain Firstly, we assume that Id(I) = (0) where z ∈ I. Since I is a nonzero ideal of R and R is a prime ring, so it implies that d(I) = (0).…”

“…During the time (1940)(1941)(1942)(1943)(1944)(1945)(1946)(1947)(1948)(1949)(1950), when the general structure theory of rings was in progress, a significant amount of work was done by Jacobson, Herstein, Bell, Kezlan, Abu-Khuzam (see [19] and references therein) on certain polynomial constraints that force a ring to be commutative. Nowadays, there has been an ongoing interest in the investigation of polynomial constrains involving various types of derivations of rings; such constrains are usually known as differential identities.…”

Posner's second theorem with two variableσ-derivations

“…Indeed, he proved that if d is a non-zero derivation of R such that [[d(a), a], b] = 0 for all a, b ∈ R, then R is commutative. During a few recent decades, this result has made a great deal of excitement among the mathematician to investigate the relationship between the commutativity of the ring R and certain special types of maps on R (see [1,10,11]).…”

Centralizing mappings, Morita context and generalized (α, β)-derivations

“…Inspired from this result, several techniques are developed to investigate conditions under which a ring becomes commutative, for instance, generalizing Herstein's conditions, using restrictions on polynomials, introducing derivations and generalized derivations on rings, looking at special properties for rings, etc. For more details and references see the review article [2]. One can also achieve this goal by comparing two rings and imposing conditions on them.…”