i am doing proof of wedderburn artin theorem from T Y Lam but the fact used in proof is decomposing semisimple Ring R as FINITE direct sum of minimal left ideals, but in in definition it is said to be direct sum of a family of minimal ideals, and explaination is as 1 belongs to R so sum will be FINITE. I don't get it, i know direct sum means only finitely many non zero elements but why 1 belong to R forces only finite number of minimum ideals in decomposition of semisimple ring R. i know answer to it is very short but its just not clicking. any help or hint? thanks
See theorem 3.4 on p. 16 here. In short, a partition of unity picks out a sufficient finite set of minimal left ideal summands to span the ring.