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.

In any case, when R is a

prime ring, all that we need here about this object is that

In [3], 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.

1] If R is a 2-torsion free semi

prime ring and a, b are elements in R then the following are equivalent.

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 [3] 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][.

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.

Let R be a

prime ring of characteristic different from 2, d a non-zero derivation of R and I a non-zero right ideal of R such that [d(x)[x.

They include discussions of the Galois map and its induced maps, when direct sums of modules inherit certain properties,

prime rings with left derivations, some commutativity theorems concerning additive mappings and derivations on semi-

prime rings, a short proof that continuous modules are clean, and imprimitive regular action in the ring of integers modulo n.