Hopfian and co-Hopfian objects
The author in this paper has made an exhaustive study of Hopficity and co-Hopficity in modules, rings, topological spaces and manifolds. The main results are the following ones. (i) A ring \(A\) is Hopfian in \(A\)-mod if and only if it is Hopfian in mod-\(A\). (ii) If a ring \(A\) is co-Hopfian in \(A\)-mod then it is Hopfian in \(A\)-mod. (iii) If \(A\) is a ring and \(n \geq 1\), then (a) \(M_ n(A)\) Hopfian (resp. co-Hopfian) as a ring \(\Rightarrow A\) Hopfian (resp. co-Hopfian) as a ring. (b) \(M_ n(A)\) Hopfian (resp. co-Hopfian) in \(M_ n(A)\)-mod \(\Rightarrow A\) Hopfian (resp. co-Hopfian) in \(A\)-mod. (c) \(M_ n(A)\) Hopfian (resp. co-Hopfian) in \(A\text{-mod}\Rightarrow A\) Hopfian (resp. co-Hopfian) in \(A\)-mod. (iv) If \(A\) is a ring and \(G\) a group then \(A[G]\) Hopfian (resp. co- Hopfian) as a ring \(\Rightarrow A\) Hopfian (resp. co-Hopfian) as a ring and \(G\) Hopfian (resp. co-Hopfian) as a group. (v) If \(M \in A\)-mod, \(M\) Hopfian in \(A\text{-mod} \Leftrightarrow M[X]\) Hopfian in \(A[X]\text{- mod}\Leftrightarrow MX\) Hopfian in \(AX\)-mod. (vi) Let \(A\) be a commutative ring. If \(A\) is co-Hopfian as \(A\)-module then, for each \(n \geq 1\), \(A^ n\) co-Hopfian as \(A\)-module and \(M_ n(A)\) is co-Hopfian in both \(M_ n(A)\)-mod and mod-\(M_ n(A)\). (vii) A Boolean ring \(A\) is Hopfian (resp. co-Hopfian) as a ring if and only if the associated Boolean space \(X_ A\) is co-Hopfian (resp. Hopfian) in the category Top. (viii) If \(X\) is a compact Hausdorff space then the algebra \(C(X)\) of all continuous real-valued functions on \(X\) is Hopfian (resp. co-Hopfian) as \(R\)-algebra if and only if \(X\) is co-Hopfian (resp. Hopfian) in Top. (ix) Any compact manifold without boundary is co-Hopfian in Top but any compact manifold with nonempty boundary is never co-Hopfian in Top. The author concludes the paper with some related results, counterexamples and open problems.
scientific article; zbMATH DE number 94222