Fully idempotent rings every maximal left ideal of which is an ideal (Q1203885)

The author constructs a counterexample to settle simultaneously the following questions all in the negative which have been raised by \textit{R. Yue Chi Ming} in 1987 and 1989; (1) Is a ring \(R\) von Neumann regular if every maximal essential left ideal of \(R\) is an ideal and every ideal of \(R\) is idempotent? [Port. Math. 44, 101-112 (1987; Zbl 0635.16011)]. (2) Is a ring \(R\) strongly regular if every maximal left ideal of \(R\) is an ideal and every ideal of \(R\) is idempotent? [Bull. Soc. Math. Belg., Sér. B 41, 129-138 (1989; Zbl 0672.16011)]. Let \(S\) be a simple radical ring in the sense of Jacobson such that \(s+s=0\) for all \(s\) in \(S\), and let \(R\) denote the set of pairs \((\overline{n},s)\) where \(\overline{n}=n+(2)\) in the field \(Z/(2)\) and \(s\) in \(S\). Define addition by \((\overline{n},s)+(\overline{m},t)=(\overline{n}+\overline{m},s+t)\), and define multiplication by \((\overline{n},s)(\overline{m},t)=(\overline{n} \overline{m},nt+ms+st)\), it is easy to see that \(R\) is a ring with identity \((\overline{1},0)\). The author proves the following results: (1) \(R\) is a fully idempotent ring; (2) \(R\) is a prime ring; (3) \(R\) is a local ring; (4) Every maximal left ideal of \(R\) is an ideal of \(R\), in particular, every maximal essential left ideal of \(R\) is an ideal of \(R\); (5) Every maximal right ideal of \(R\) is an ideal of \(R\), in particular, every maximal essential right ideal of \(R\) is an ideal of \(R\); (6) \(J(R)=\{(\overline{0},s)\mid s \text{ in }S\}\neq 0\). Hence \(R\) is not von Neumann regular. It is noteworthy that the ring \(R\) is useful in ring theory.