Arnautov, V. I. Properties of accessible subrings of topological rings when taking quotient rings. (English) Zbl 1184.16047 Bul. Acad. Ştiinţe Repub. Mold., Mat. 2007, No. 2(54), 4-18 (2007). Topological rings are not assumed to be associative or Hausdorff. A continuous isomorphism \(f\colon R\to R'\) is called ‘semitopological’ provided there exists a topological ring \(S\) containing \(R\) as an ideal and an open continuous homomorphism \(g\colon S\to R'\) such that \(g\upharpoonright_R=f\) [V. I. Arnautov, Mat. Issled. 4, No. 2(12), 3-16 (1969; Zbl 0232.16031)]. A subring \(S\) of a ring \(R\) is said to be a ‘subideal’ if there is a finite chain \(S=R_m\leqslant R_{m-1}\leqslant\cdots\leqslant R_0=R\), such that \(R_i\) is an ideal of \(R_{i-1}\), \(i=1,\dots,n\). The author considers the following question: When is the superposition of semitopological isomorphisms semitopological? The answer is given in Theorem 3: A continuous isomorphism \(f\colon R\to R'\) is semitopological \(\Leftrightarrow\) \(R\) is a subideal of a topological ring \(S\) and there exists an open continuous homomorphism \(g\colon S\to R'\) such that \(g\upharpoonright_R=f\). It is proved (Theorem 4) that every continuous isomorphism of a nilpotent in the abstract sense topological ring on another topological ring is a superposition of semitopological isomorphisms. Reviewer: Mihail I. Ursul (Oradea) MSC: 16W80 Topological and ordered rings and modules 54H13 Topological fields, rings, etc. (topological aspects) 54C10 Special maps on topological spaces (open, closed, perfect, etc.) Keywords:topological rings; fundamental systems of neighborhoods of zero; continuous isomorphisms; semitopological isomorphisms; accessible subrings Citations:Zbl 0232.16031 PDFBibTeX XMLCite \textit{V. I. Arnautov}, Bul. Acad. Științe Repub. Mold., Mat. 2007, No. 2(54), 4--18 (2007; Zbl 1184.16047)