Howard E. Bell

Near-rings in which each element is a power of itself

Let R denote a near-ring such that for each x ∈ R , there exists an integer n ( x ) > 1 for which x n(x) = x . We show that the additive group of R is commutative if 0.x; = 0 for all x ∈ R and every non-trivial homomorphic image R¯ of R contains a non-zero idempotent e commuting multiplicatively with all elements of R¯. As the major consequence, we obtain the result that if R is distributively-generated, then R is a ring – a generalization of a recent theorem of Ligh on boolean near-rings.

Remarks on derivations on semiprime rings.

We prove that a semiprime ring R must be commutative if it admits a derivation d such that (i) xy

On Derivations in Near-Rings, II

Let N denote a 3–prime zero-symmetric left near-ring, and let d be a nonzero derivation on N. Let U ≠ φ be a nonzero subset of N such that (i) U N ⊆ U or (ii) N U ⊆ U. We prove that N must be a commutative ring if one of the following holds: (a) U satisfies one of (i) and (ii), and d(U) is multiplicatively central; (b) U satisfies both of (i) and (ii), d 2 ≠ 0, and [d(U), d(U)] = {0}. Some related results are also given.

On derivations and commutativity in semiprime rings

Let R be a ring, Z its center, U a nonzero left ideal, and D:R → R a derivation. We show that if R is semiprime with suitably-restricted additive torsion, then R must contain nonzero central ideals if one of the following holds: (i) [x, [x, D(x)]] ∊ Z for all x ∊ U; (ii) for a fixed positive integer n, [xn, D(x)] ∊ Z for all x ∊ U

On Prime Near-Rings with Generalized Derivation

Let 𝑁 be a 3-prime 2-torsion-free zero-symmetric left near-ring with multiplicative center 𝑍. We prove that if 𝑁 admits a nonzero generalized derivation 𝑓 such that 𝑓(𝑁)⊆𝑍, then 𝑁 is a commutative ring. We also discuss some related properties.

A Near-Commutativity Property for Rings

We revisit rings with the property that |A2| ≤ 3 for each 2-subset A. We then investigate commutativity of rings R with the property that for each pair a, b of noncommuting elements of R, there exists an integer n > 1 for which an = bn.

Rings with finitely many nilpotent elements

It is well-known that a ring with no nonzero nilpotent elements - a so-called reduced ring - is a subdirect product of domains. Moreover, as we have recently shown [2], a prime ring with only finitely many nilpotent elements is either a domain or is finite. In view of these results, it is natural to ask what can be said in general about rings with only finitefy many nilpotent elements. A crucial property of such rings is that they contain no infinite zero subrings, hence we are led to consider rings with this property also. Our principal result is that for any ring R with only finitely many nilpotent elements is a direct sum of a reduced ring and a finite ring, where p(R) denotes the prime radical of R. One consequence is a finiteness theorem for periodic rings; another is the rather surprising result that every ring with infinitely many nilpotent elements has an infinite zero subring.

Some conditions for finiteness and commutativity of rings

We present several new sufficient conditions for a ring to be finite; we give two conditions which for periodic rings R imply that R nst be either finite or cumutative; and we study cumutativity in rings with only finitely many non-central subrings.

Some commutativity results for rings

It is proved that certain rings satisfying generalized-commutator constraints of the form [ x m , y n , y n , …, y n ] = 0 must have nil commutator ideal.

ON THE COMMUTATIVITY OF PRIME RINGS WITH DERIVATION

Abstract We study commutativity of prime rings R such that d(S) is central, where d is a non-zero derivation and S a suitably-chosen subset of R.