On rings in the join of associative and commutative (Q1194051)

For ring elements \(a,b,c\) let \([a,b]:=ab-ba\) and \((a,b,c):=(ab)c-a(bc)\). The authors study rings \(R\) such that \(([a,b],c,d)=0\) and \([(a,b,a),c]=0\) for all \(a,b,c,d\in R\). By skillful constructions of ideals they prove: If \(R\) is semiprime then \(R\) is isomorphic to a subdirect sum of a semiprime associative ring and a semiprime commutative ring. If \(R\) is prime then \(R\) is associative or commutative.