On Hopfian rings (Q1307410)

A ring \(R\) (associative with \(1\neq 0\)) is called Hopfian if every surjective ring homomorphism \(f\colon R\to R\) is an isomorphism. A (Hausdorff) topological space is called co-Hopfian, if every injective continuous selfmap \(f\colon X\to X\) is a homeomorphism. The main results of the paper are as follows. If \(X\) is an infinite compact totally disconnected co-Hopfian space, then the Boolean ring of clopen subsets of \(X\) is a commutative non-Noetherian Hopfian ring. If \(R\) is a Boolean Hopfian ring then the polynomial ring \(R[T]\) is Hopfian. If \(R\) and \(S\) are Hopfian rings such that \(S\) has no non-trivial central idempotents and it is not a homomorphic image of \(R\), then \(R\times S\) is Hopfian.