# Universal Property of $M[x]$, where $M$ is an $R$-module.

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).

• Without any further structure, $M[x] \cong \bigoplus_{i=1}^\infty M$ is simply a countable direct sum of $M$'s. So it has the corresponding universal property, but this is probably not so interesting. You could turn $M[x]$ into an $R[x]$-module. Then $M[x] \cong R[x] \otimes_R M$. – moonlight Sep 4 '15 at 6:59

We can regard $M[x]$ as an $R$-algebra rather than a $R$-module. Change the word 'module' to 'algebra' in your description, then you get the universal property of $M[x]$.