# Precise definition of direct summand of a module

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?

• A submodule $N$ is a direct summand of $M$ if there exists a submodule $N'$ of $M$ such that $M$ is the (internal) direct sum $N\oplus N'$, i.e., for every $m\in M$ there exists unique $(n,n')\in N\times N'$ such that $m=n+n'$. Apr 11 at 14:53
• Your definition is good. To be extremely precise, one should probably distinguish between internal direct summands and external direct summands but since this is usually/always clear from the context, it isn't done. In your case, yes, $2 \mathbb{Z}$ is an (external) direct summand but I wouldn't phrase it that way as the $2$ can be confusing and the important thing here is rather that $2 \mathbb{Z}$ is embedded via $2 \mathbb{Z} \cong \mathbb{Z}$ in $\mathbb{Z}$. Apr 12 at 6:10

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.
• @Anupam That depends on the context, for most purposes it suffices to say that it is any module $A$ for which there exists a module $C$ such that $B \cong A \oplus C$. (And certainly, one could then identify $A$ with a submodule of $B$.) Ok, I just saw that this was exactly your textbook definition as well. Apr 12 at 5:39