On linearly compact rings (Q1113988)

For a topological ring \(R\) and infinite cardinal \(\alpha\) one defines \(\text{T-card}\, R<\alpha\), if the cardinality of \(R/U\) is less than \(\alpha\) for each open right ideal \(U\). If \(R\) is strictly linearly compact with an ideal basis filter, then \(R\) has two uniquely determined closed ideals \(S(\alpha)\) and \(S^*(\alpha)\) such that \(S(\alpha)\) is the largest closed right ideal of \(R\) with \(\text{T-card}\,S(\alpha)<\alpha\), and \(S^*(\alpha)\) is the smallest closed ideal with \(\text{T-card}\,R/S^*(\alpha)<\alpha\). Several consequences are deduced, e.g. if \(R\) is right and left strictly linearly compact, then \(R=S(\alpha)\oplus S^*(\alpha)\) in algebraic and topological sense.