Precise definition of direct summand of a module

Question

Can someone give a precise definition of a direct summand of an $R$-module $M$. (You can assume $R$ is commutative with unity).

Here is what I thought till date:

"We say an $R$-module $N$ is a direct summand of $M$ if there exists an $R$-module $N'$ such that $M$ is isomorphic to $N \oplus N'$".

But while going through some articles on commutative algebra/ homological algebra, I don't think this is taken as a defintion.

For example, in my definition, $2\mathbb{Z}$ is a direct summand of $\mathbb{Z}$. But is it really so according to standard literature? Can someone point me to a definition of direct summand in some popular textbook?

Answer 1

A submodule $A$ of $B$ is a direct summand if there exists a $C$ such that $B\cong A\oplus C$ where the canonical map of the first summand is the inclusion map.