Additivity of maps on $\mathbb{N}\times\mathbb{N}$ matrices

Abstract

It was recently proved that if a map defined on the ring of all $n\times n$ matrices satisfies the condition $f(x+y)=f(x)+f(y)$ for all invertible $x$ and $y$, then $f$ is additive. In the present paper, it is proved that the same result holds for the ring of $\mathbb{N}\times\mathbb{N}$ matrices with a finite number of nonzero entries in each column defined over a field of at least three elements.