Automorphisms of subrings of \(\text{RFM}(R)\) (Q1841999)

Let \(R\) be an associative ring with identity element. \(\text{RFM}(R)\) denotes the ring of row-finite matrices over \(R\) and \(FC(R)\) its twosided ideal consisting of all the matrices with a finite number of nonzero columns. In this paper, we consider subrings \(\Sigma\) of \(\text{RFM}(R)\) such that \(FC(R) \subseteq \Sigma\), and study some properties of the automorphisms \(\sigma\) of such subrings. Most of these properties are related to the structure of the endomorphism ring of the left \(R\)-module \(R^{(\mathbb{N})} \sigma(e)\), where \(e\) is the matrix with 1 in the left upper corner, 0 elsewhere. In particular, it is shown that this endomorphism ring is not necessarily isomorphic to the ring \(R\).