Ring of locally finite matrices

Question

Let $I$ be a countable set. A matrix $A=(a_{ij})_{i,j \in I}$ is locally finite if for every $i \in I$, the families $(a_{ij})_{j\in I}$ and $(a_{ji})_{j\in I}$ have finite support . Denote by $M_{I}(\mathbb{Z})$ the ring of locally finite matrices with integral entries indexed by $I \times I$. My question is, how is this ring defined if $|I|= \infty$? I guess addition is just the usual matrix addition but what about the multiplication?

Answer 1

As usual: $(A.B)_{ij}=\displaystyle\sum_{k\in I} a_{ik}b_{kj}$. Since $A$ has finite support for each $i\in I$, $a_{ik}=0$ except for a finite number of indices $k$, so the sum is finite.