Structures of rings satisfying certain conditions
Let \(R\) be an associative ring and \(f(t)\in tZ[t]\). The author considers the following conditions: \(\text{(H)}_f\) For each \(x,y\in R\), there exists \(k >0\) such that \([f(x),f(y)]_k=0\); (S) For each \(a,b\in R\), there exists a polynomial \(f(x,y)\) from the commutator ideal of a free 2- generated ring such that \([a,b]=f(a,b)\) and each monomial of \(f(x,y)\) is of length \(\geq 3\). \(Q(n)\) If \(a,b \in R\) and \(n[a,b]=0\) then \([a,b]=0\). He investigates the structure of rings satisfying some of these conditions. In particular, the following main results are proved: 1) Let \(R\) be a ring with unity satisfying the conditions \((H)_f\), \((S)\), \(Q(f(1))\) where \(f(t)\) is a comonic polynomial such that \(f(1) \neq 0\). Then \(R\) is a subdirect product of rings each of which has one of the following types: i) commutative ring; ii) \(M_\sigma(K)=\bigl\{\left(\begin{smallmatrix} a &b \\ 0 &\sigma(a)\end{smallmatrix}\right);\;a,b\in K=\text{GF}(p^n),\;\sigma\in\text{Aut }K,\;\sigma\neq 1\bigr\}\), and \((p,f(1))=1\), \(|K|/ |K^\sigma|\leq \text{deg }f(t)\). 2) Let \(R\) satisfy the conditions \((H)_f\), \((S)\) and \(Q(f(1)!)\) where \(f(t)\) is a comonic polynomial with non-negative coefficients. Then \(R\) is a subdirect product of rings each of which has one of the following types: i) commutative ring; ii) \(M_\sigma (K)\), where \((p,f(1)!)=1\), \(|K|/|K^\sigma |\leq\text{deg }f(t)\), \((|K|- 1/|K^\sigma|-1)|f'(1)\).
scientific article; zbMATH DE number 792564