The young bulls were again on fire as the prime ring
on a Thursday remains a hot spot our vendors of feeding bulls continue to reap the benefits, top price was a 14 month old British Blue cross from T L Holmes & Son, White House Farm, Craghead which sold for PS1220 to J A Matten & Sons, Avenue Grange, Thirsk.
Bresar and Zalar () proved that if R is a prime ring
and [phi] is an additive mapping on R such that [phi]([A.sub.2]) = [phi](A)A (resp., [phi]([A.sub.2]) = A[phi](A)) for any A [member of] R, then [phi] is a left (resp., a right) centralizer.
In any case, when R is a prime ring
, all that we need here about this object is that
In , Ashraf and Rehman established that a prime ring
R with a nonzero ideal I must be commutative, if R admits a nonzero derivation d satisfying d(xy) + xy [member of] Z(R) for all x,y [member of] I or d(xy) - xy [member of] Z(R) for all x,y [member of] I.
A well known result of Posner (19) states that for a non-zero derivation d of a prime ring
R, if [[d(x), x], y] = 0 for all x, y [member of] R, then R is commutative.
 If R is a 2-torsion free semi prime ring
and a, b are elements in R then the following are equivalent.
However, a classical result of Herstein (, Theorem 3.3) shows that a Jordan derivation of a 2-torsion free prime ring
is a derivation.
The impressive black steer will be destined to hopefully one day return through our prime ring
as he was purchased by regular supporter Ricky Alder of M F Hall, Woodhill Farm, Ponteland for PS1220.
Throughout the paper unless specifically stated, R always denotes a prime ring
with center Z(R) and extended centroid C, right Utumi quotient ring U.
In  Bell and Kappe proved that if d is a derivation of a prime ring
R which acts as a homomorphism or as an anti-homomorphism on a nonzero right ideal I of R, then d = 0 on R.
Let R be a prime ring
of characteristic [not equal to] 2 with right quotient ring U and extended centroid C, g [not equal to] 0 a generalized derivation of R, L a non-central Lie ideal of R and n [greater than or equal to] 1 such that [g(u), u][.sup.n] = 0, for all u [member of] L.
"The breed has become hugely popular in recent years, much of which can be attributed to prime lamb prices which command a premium in the prime ring