A class of generalized radical rings
For the ring \(R\) and \(A,B\subseteq R\), \(a\in R\) are denoted: \(A-B=\{a- b\mid a\in A\), \(b\in B\}\), \(G^{(a)}=\{x\in R\mid xa=ax=x\}\), \(E=\{e\in R\mid e^ 2=e\neq 0\}\). \(R\) is called DJ-ring if for each \(a\in R\) the set \(a-G^{(a)}\) contains at least one quasi-regular element and at most one two-sided nonquasi-regular element. The following statements on \(R\) are equivalent: (i) \(R\) is DJ-ring, \(R\neq J(R)\) and there is an \(e\in E\) such that \(E-e\in J(R)\); (ii) \(R=eRe+eR(1-e)+(1-e)Re\), where \(0\neq e=e^ 2\in R\), \(eR(1-e)+(1-e)Re\) is a subring of \(R\) and \(eRe\) is a division ring. A lattice isomorphism between the lattices \(\Sigma=\{S \vartriangleleft R\mid S\nsubseteq J(R)\}\) and \(\Sigma'=\{T\vartriangleleft R\mid T=J(R)\), \(\exists u\in E\), \(uJ(R)\subseteq T\), \(J(R)u\subseteq T\}\) is established.
scientific article; zbMATH DE number 148465