Why is this subset finite?

Question

We are given that $R$ is a ring with identity, and that $R$ is left semisimple, i.e. $R$ can be decomposed into a sum of minimal left ideals of $R$ ($R=\bigoplus_{n\in S} I_n$). What I'm confused about is why the cardinality of the set of projections of $1$ into each $I_n$ with such a projection being nonzero is finite, i.e. why we can say $$1=1_{n_1}+\cdots+1_{n_k}$$ For some $k\in \mathbb{N}$ and $1_{n_i}\in I_{n_i}$. Here is the text I am looking at. It is from Knapp's Advanced Algebra, Chapter 2

Answer 1

That’s just what a direct sum is. Any element of a direct sum only has finitely many non-zero components. More symbolically, an element of $\oplus_{n\in S}I_i$ is a tuple $(a_n)_{n\in S}$ with $a_n\in I_n$ such that $a_n=0$ for all but finitely many $n$.