On radicals of rings which are sums of two subrings
Rings which are sums of two subrings have been studied in many papers. In this paper, the authors study the radicals of such rings. They obtain new results concerning the open questions of whether a ring which is a sum of two prime radical subrings must be prime radical. The authors use the symbol ``\(A\in S\)'' or ``\(A\) is an \(S\)-ring'' to mean a ring \(A\) has property \(S\). A radical \(S\) is hereditary if whenever a ring \(A\in S\) and \(I\) is an ideal of \(A\), then \(I \in S\). Hereditary radicals containing the prime radical \(\beta\) are called supernilpotent. A radical \(S\) is right strong if for every ring \(A\), \(S(A)\) contains all right \(S\)-ideals of \(A\). Given a ring \(A\), denote the left-annihilator of \(A\) by \(l(A)\). Throughout this paper \(R\) is a ring and \(R_1\), \(R_2\) are subrings of \(R\) such that \(R=R_1+R_2\). The authors prove Theorem 1: If \(R_1\in\beta\) and \(R_2\in S\), where \(S\) is a right strong and supernilpotent radical, then \(l(R_1)\subset S(R)\). They give an example to show the assumption that \(R_1\in\beta\) cannot be omitted from this theorem. It is known that if a radical \(S\) containing \(\beta\) is left or right strong and left or right hereditary, then it is both left and right strong as well as left and right hereditary. Radicals of that type are called \(N\)-radicals. The problem whether the nil radical is an \(N\)-radical is still open; it is called Koethe's problem. The authors' Theorem 1 enables them to prove if \(R_1\) is a nilpotent ring and \(S\) is an \(N\)-radical then \(R\in S\). A ring \(A\) is left \(T\)-nilpotent if for every non-zero homomorphic image \(A'\) of \(A\), \(l(A')\neq 0\). The authors prove that if \(R_1\) is a nil PI ring and \(R_2\in S\), where \(S\) is an \(N\)-radical, then \(R\in S\). This theorem enables them to prove the following corollaries: (i) Let \(S\) be an \(N\)-radical. If \(R_1\) is a nil ring of bounded index and \(R_2\in S\), then \(R\in S\). (ii) If \(R_1\) is a commutative nil ring and \(R_2\) is a left or right \(T\)-nilpotent ring, then \(R\) is a prime radical ring. (iii) If \(R_1\) is a commutative nil ring and \(R_2\) is a Wedderburn radical ring, then \(R\) is a prime radical ring. (iv) If \(R_1\) and \(R_2\) are nil rings of bounded index then so is \(R\). (The proof of this corollary does not give any relation between the indices of \(R_1\), \(R_2\), and that of \(R\). The authors have been unable to find a proof which would give such a relation).
scientific article; zbMATH DE number 937064