Let \(R\) be an associative ring \(R\). To denote that \(L\) is a left ideal of \(R\), the authors write \(L<_lR\). Let \(A(R)=\{\sum L<_lR:L\text{ is nil}\}\) be the Andrunakievich ideal of \(R\). If \(R=A(R)\), the ring \(R\) is called an \(A\)-ring.
In this paper the authors study properties of two-sided and one-sided ideals of \(A\)-rings. In particular, they describe relations between \(A(R)\) and \(A(I)\), where \(I\) is an ideal or left ideal of \(R\).
The famous Köthe problem asks whether the nil radical of a ring \(R\) contains all the nil left ideals (equivalently, right) ideals of \(R\). The authors prove that Köthe’s problem has a positive solution if and only if one-sided ideals of \(A\)-rings are again \(A\)-rings.
Another related open problem asks whether annihilators of elements of non-zero \(A\)-rings are non-zero. The authors present some results on this problem. They prove, for example, that the left and the right annihilator of every nonzero element of a ring whose every left ideal is an \(A\)-ring is nonzero.