I am reading Dummit & Foote Chapter 10 and stuck on this exercise ($R$ is a ring with $1$ and $M$ is a left $R$-module):
I am unsure about how to proceed in (i) => (ii). I assume the direct sum in (i) means external direct sum (as the problem itself is essentially defining internal direct sum). We are given an isomorphism, but we don't know whether it's the natural one. How can we use this isomorphism to prove (ii)?