I have stumbled upon two different definitions of a submodule and I cant see why they are equivalent, let me state them ($M$ is a module and $R$ a commutative ring):
First Definition: A set $N \subset M$ is called a submodule of $M$ if $$an + bm \in N$$ for all $a,b \in R$ and all $m,n \in N$
Second definition: A set $N \subset M$ is called a submodule of $M$ if (i) $m+n \in N$ for all $m,n \in N$ and (ii) $an \in N$ for all $a \in R$ and $n \in N$.
However I cant see they the definitions are equivalent for instance; if the second definition holds then the first holds, but the converse is not true? If the first definitions holds then we can choose $a=b=1$ thus (i) in the second definition holds but how can I get (ii) to hold in the second definition? More precisely: The fact that $an+bm \in N$ does not imply that $an \in N$, is this correct or am I missing something?
There must be a $0$ in the ring $A$. Let $b = 0$ in $an + bm \in N$; then $an + 0m = an \in N$.