Let $R$ be a ring (commutative with identity) and $M$ be an $R$-module. We define $M[x]$ as the set of all the formal sums $a_0+a_1x+\cdots +a_kx^k$ with $a_i\in M$. We give it an $R$-module structure by defining addition and $R$-scaling in an obvious way.
Does $M[x]$ have a universal property?
Inspired from the universal property of $R[x]$, I was tempted to think that $M[x]$ has the following universal property: Let $i:M\to M[x]$ be the natural injection. Given any $R$-module homomorphism $g:M\to N$ and given an element $n\in N$, there is a unique $R$-module homomorphism $F:M[x]\to N$ such that $F\circ i=f$ and $F(x)=n$.
But this makes no sense since we are not allowed to multiply things in $N$.
So what is the universal property (if there is one).