I'm working on the following question from Chapter 8 of Algebra: Concrete & Abstract.
Let $R$ denote the set of infinite-by-infinite, row- and column-finite matrices with complex entries. That is, a matrix is in $R$ if and only if each row and each column has only finitely many non-zero entries. Show that $R$ is a non-commutative ring with identity, and that $R \cong R \oplus R$.
My difficulty is with the last part. I was trying to do the following. Let $R_e$ be the submodule of $R$ where all the non-zero entries are in even columns. Similarly let $R_o$ be the submodule of $R$ where all the non-zero entries are in odd columns. We note that $R_o \cong R \cong R_e$. I want to say that since $R \cong R_o \oplus R_e$ as abelian groups, this carries over to a ring isomorphism. I'm not sure if this is true. I would appreciate any comments about whether I'm in the right direction or not.