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On Johns rings. (English) Zbl 1189.16020

Summary: A ring \(R\) is called right Johns if \(R\) is a right Noetherian ring such that every right ideal \(A\) of \(R\) is an annihilator (i.e. \(rl(A)=A\) for all right ideals \(A\) of \(R\)). \(R\) is called strongly right Johns if every matrix ring over \(R\) is right Johns. In this paper it is shown that every right Johns ring is a cogenerator and hence quasi-Frobenius. By using this result we show that every strongly right Johns ring is quasi-Frobenius without any extra condition.

MSC:

16P40 Noetherian rings and modules (associative rings and algebras)
16L60 Quasi-Frobenius rings
16S50 Endomorphism rings; matrix rings
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