Rings which are sums of two subrings
Suppose that an associative ring \(R\) has the form \(R=R_1+R_2\), where \(R_1\) and \(R_2\) are subrings of \(R\). One can ask what properties of \(R\) can be derived if \(R_1\) and \(R_2\) satisfy certain conditions. The main studies of rings which are sums of two subrings concern radicals and polynomial identities. The authors also concentrate on these topics in this paper. Given a ring \(A\), define the left hyperannihilator \(l(A)\) of \(A\) to be the union \(\bigcup_{\alpha\geq 0}l_\alpha(A)\) where \(l_0(A)=0\) and for \(\alpha\geq 1\) \[ l_\alpha(A)=\bigl\{x\in A\mid xA\subseteq\bigcup_{\beta<\alpha}l_\beta(A)\bigr\}. \] Similarly define \(r(A)\), the right hyperannihilator of \(A\). The intersection \(N(A)=l(A)\cap r(A)\) is called the hyperannihilator of \(A\). In a paper written in 1964, O. H. Kegel asked whether at least one of \(N(R_1)\) or \(N(R_2)\) is contained in \(\beta(R)\), the prime radical of \(R\). The authors give an example which answers Kegel's question in the negative. Their example also shows that there are no direct relations between radicals of \(R\) and those of \(R_1\) and \(R_2\). It has been previously noted that the field of rational numbers is a sum of two Jacobson radical subrings. In the second part of this paper the authors characterize all fields of that type. They prove that for a field \(F\) the following are equivalent: 1. \(F=A+B\) for some Jacobson radical rings \(A\) and \(B\); 2. \(F=A+B\) for some proper subrings \(A\) and \(B\) of \(F\); 3. \(\text{ch }F=0\) or \(\text{tr}_{F_0}F>0\), where \(F_0\) is the prime subfield of \(F\). A descending chain \(I_1\supseteq I_2\supseteq\cdots\) of ideals of a ring \(A\) is said to absorb \(A\) if for every natural number \(n\) one has \(AI_n+I_nA\subseteq I_{n+1}\). O. H. Kegel asked whether the existence of properly descending absorbing chains of ideals of \(R_1\) and \(R_2\) is sufficient to force the ring \(R\) to be non-simple. The authors show that the answer is negative. The authors also answer an open question of Yu. Bakhturin and A. Giambruno. Namely, they prove that if \(R_1\) is a nil ring of bounded index and \(R_2\) is a ring satisfying a polynomial identity, then \(R\) satisfies a polynomial identity. The question of whether there exists a simple nil ring is still open. In the final section of this paper, the authors show that to construct a simple ring which is a sum of two nil subrings is as hard as to construct a simple nil ring itself.
scientific article; zbMATH DE number 1308264
scientific article