It is well known that weak regularity is equivalent to regularity and biregularity for left Artinian rings. In this paper the authors prove the following result: for a ring \(R\) satisfying the ACC on right annihilators, if \(R\) is left weakly regular then \(R\) is biregular, and that \(R\) is left weakly regular if and only if \(R\) is a direct product of a finite number of simple rings. Next the authors study maximality of strongly prime ideals, showing that a reduced ring \(R\) is weakly regular if and only if \(R\) is left weakly regular if and only if \(R\) is left weakly \(\pi\)-regular if and only if every strongly prime ideal of \(R\) is maximal.