Ideals contained in subrings (Q2705774)

Let \(S\) be a ring and \(R\) a subring. By the index of \(R\) in \(S\), we shall mean the index of \((R,+)\) in \((S,+)\). A useful result of \textit{J. Lewin} [see J. Algebra 5, 84-88 (1967; Zbl 0143.05303)] says that if \(R\) is a subring of finite index in \(S\), then \(R\) contains an ideal of \(S\) which is also of finite index. The authors provide some extensions of this result, and apply them to prime rings.