Alladi Ramakrishnan Hall

Jordan derivations and related maps in rings

Shakir Ali

Aligarh Muslim University

Let R be an associative ring. For any x, y ∈ R, as usual the symbols x ◦ y
and [x, y] will denote the anti-commutator xy + yx and commutator xy − yx and
called Jordan product and Lie product, respectively. Recall that a map f of a
ring R into itself is said to be additive if f (x + y) = f (x) + f (y) for all x, y ∈ R.
An additive map d : R → R is called a derivation if d(xy) = d(x)y + xd(y) holds
for all x, y ∈ R. An additive map d : R → R is called a Jordan derivation if
d(x2 ) = d(x)x + xd(x) holds for all x ∈ R. An additive map x → x∗ of R into
itself is called an involution if (i) (xy)∗ = y ∗ x∗ and (ii) (x∗ )∗ = x hold for all
x, y ∈ R. A ring equipped with an involution is known as a ring with involution
or a ∗-ring. An additive map d : R → R is called a Jordan ∗-derivation if
d(x2 ) = d(x)x∗ + xd(x) holds for all x ∈ R.

In this talk, I will review some recent results of myself and collaborations
in certain class rings involving these mappings. Moreover, some examples and
counter examples will be discussed for questions raised naturally.