Palestine Journal of Mathematics
Vol. 3(Spec 1) (2014) , 445–448 © Palestine Polytechnic University-PPU 2014
On α−
∗
Centralizers of Semiprime Rings With Involution
Öznur Gölba¸ sı and Ahu Kadriye Gürcan
Dedicated to Patrick Smith and John Clark on the occasion of their 70th birthdays.
Communicated by Ayman Badawi
MSC 2010 Classifications: 16W10, 16N60.
Keywords and phrases: Semiprime ring, centralizer, α-
*
centralizer.
Abstract. Let R be a semiprime ring equipped with an involution ∗ and α be an epimorphism
of R. In this paper, we prove that an additive mapping T : R → R is a Jordan α−
*
centralizer if
the following holds:
2T (xyx)= T (x)α(y
*
)α(x
*
)+ α(x
*
)α(y
*
)T (x), for all x, y ∈ R.
1 Introduction
Throughout, R will represent an associative ring with center Z . Recall that a ring R is prime
if xRy = 0 implies x = 0 or y = 0, and semiprime if xRx = 0 implies x = 0. An additive
mapping x → x
*
satisfying (xy)
*
= y
*
x
*
and (x
*
)
*
= x for all x, y ∈ R is called an involution
and R is called a ∗-ring.
According B. Zalar [8], an additive mapping T : R → R is called a left (resp. right) central-
izer of R if T (xy)= T (x) y (resp. T (xy)= xT (y)) holds for all x, y ∈ R. If T is both left as
well right centralizer, then it is called a centralizer. This concept appears naturally C
*
−algebras.
In ring theory it is more common to work with module homorphisms. Ring theorists would write
that T : R
R
→ R
R
is a homomorphism of a ring module R into itself instead of a left centralizer.
In case T : R → R is a centralizer, then there exists an element λ ∈ C such that T (x)= λx for
all x ∈ R and λ ∈ C, where C is the extended centroid of R.
An additive mapping T : R → R is said to be a left (resp. right) Jordan centralizer if
T
(
x
2
)
= T (x) x (resp. T
(
x
2
)
= xT (x)) holds for all x ∈ R. Zalar proved in [8] that any left
(right) Jordan centralizer on 2−torsion free semiprime ring is a left (right) centralizer. Recently,
in [1], E. Alba¸ s introduced the definition of α−centralizer of R, i. e. an additive mapping
T : R → R is called a left (resp. right) α−centralizer of R if T (xy)= T (x) α (y) (resp.
T (xy)= α (x) T (y)) holds for all x, y ∈ R, where α is an endomorphism of R. If T is left and
right α−centralizer then it is natural to call α−centralizer. Clearly every centralizer is a special
case of a α−centralizer with α = id
R
. Also, an additive mapping T : R → R associated with a
homomorphism α : R → R, if L
a
(x)= aα(x) and R
a
(x)= α(x)a for a fixed element a ∈ R
and for all x ∈ R, then L
a
is a left α−centralizer and R
a
is a right α−centralizer. Alba¸ s showed
Zalar’s result holds for α−centralizer. Considerable work has been done on this topic during the
last couple of decades (see [1-8], where further references can be found).
On the other hand, it was proved that T is a centralizer if one of the following holds
2T (x
2
)= T (x)x + xT (x),
2T (xyx)= T (x)yx + xyT (x), for all x, y ∈ R,
where T : R → R is an additive mapping respectively in [5] and [7]. These results proved for
α−centralizer in [4] and [3].
Inspired by the definition centralizer, the notion of
*
−centralizer was extended as follow:
Let R be a ring with involution ∗. An additive mapping T : R → R is called a left (resp.
right)
*
−centralizer of R if T (xy)= T (x) y
*
(resp. T (xy)= x
*
T (y)) holds for all x, y ∈ R.
An additive mapping T : R → R is said to be a left (resp. right) Jordan
*
−centralizer if
T
(
x
2
)
= T (x) x
*
(resp. T
(
x
2
)
= x
*
T (x)) holds for all x ∈ R. In [2], the authors proved
that if R is a 2−torsion free semiprime ring and T : R → R is an additive mapping such that
2T (x
2
)= T (x)α(x
*
)+ α(x
*
)T (x), for all x ∈ R, then T is a Jordan α−
*
centralizer. Motivated
this result, we will prove that an additive mapping T : R → R is a Jordan α−
*
centralizer if the
following holds:
2T (xyx)= T (x)α(y
*
)α(x
*
)+ α(x
*
)α(y
*
)T (x), for all x, y ∈ R