On \(n\)-simple rings. (Q2577733)

A simple ring \(R\) with a unit element is \(n\)-simple if for any \(x\in R\) there exist elements \(a_1,\dots,b_n\in R\) such that \(1=a_1xb_1+\cdots+a_nxb_n\). Suppose that a simple ring \(R\) with a unit element is a direct sum of \(n\) minimal right ideals, then \(R\) is \(n\)-simple but not \((n-1)\)-simple. In particular, every simple Artinian ring is \(n\)-simple for some \(n\). Every 1-simple Artinian (Noetherian) ring is a skew field.