On near-periodicity and pseudoperiodicity in rings. (Q2905987)

Let \(R\) denote a ring with center \(Z\), and let \(N\) be the set of nilpotent elements. The ring \(R\) is called periodic if for each \(x\in R\), there exist distinct positive integers \(m\) and \(n\) such that \(x^n=x^m\). Call \(R\) normal if all of its idempotents are central. The authors of the paper under review define \(R\) to be nearly periodic if for each \(x\in R\setminus(N\cup Z)\), there exist distinct positive integers \(m\) and \(n\) such that \(x^n-x^m\in N\). For distinct and fixed positive integers \(m\) and \(n\), the authors define the ring \(R\) to be \((m,n)\)-pseudoperiodic (resp. weakly \((m,n)\)-pseudoperiodic) if for each \(x\in R\setminus(N\cup Z)\) (resp. each \(x\in R\setminus(J\cup N\cup Z)\)), \(x^n-x^m\in N\cap Z\).NEWLINENEWLINE The main results of the paper under review are the following:NEWLINENEWLINE Theorem 2.4. If \(R\) is any nearly periodic ring, then \(R\) is commutative or periodic.NEWLINENEWLINE Theorem 2.5. Let \(n>1\). If \(R\) is a \((2^n-2)\)-torsionfree ring with the property that \(x^n-x\in N\) for all \(x\in R\setminus(N\cup Z)\), then \(R\) is commutative or \(R=N\).NEWLINENEWLINE Theorem 3.1. Let \(R\) be an \((n,1)\)-pseudoperiodic ring such that for all \(a\in N\) and \(x\in R\), \((n-1)[a,x]=0\) implies \([a,x]=0\). If \(Z\setminus N\neq\emptyset\), then \(R\) is commutative.NEWLINENEWLINE Theorem 3.4. If \(R\) is an \((m,n)\)-pseudoperiodic ring with \(1\) such that (*) for each \(a\in N\) and \(x\in R\), \((n-m)[a,x]=0\) implies \([a,x]=0\), then \(R\) is commutative.NEWLINENEWLINE Theorem 3.5. Let \(R\) be a normal \((m,n)\)-pseudoperiodic ring satisfying (*). If \(N\) is commutative, then \(R\) is commutative.NEWLINENEWLINE Theorem 4.3. Let \(R\) be a weakly \((m,n)\)-pseudoperiodic ring with \(1\), such that (**) for each \(a\in N\cup J\) and \(x\in R\), \((m-n)[a,x]=0\) implies \([a,x]=0\). If \(J\) is commutative, then \(R\) is commutative.NEWLINENEWLINE Theorem 4.6. Let \(R\) be a normal weakly \((m,n)\)-pseudoperiodic ring such that for all \(a\in J\) and \(x\in R\), \((m-n)[a,x]=0\) implies \([a,x]=0\). If \(J\) is commutative, then \(R\) is commutative.NEWLINENEWLINE Theorem 4.7. Let \(R\) be a weakly \((m,n)\)-pseudoperiodic ring satisfying (**). If \(J\) is commutative and \(Z\) contains an element which is not a zero divisor in \(R\), then \(R\) is commutative.