Is $\mathbb{R}[x^3]$ $ \subseteq$ $\mathbb{R}[x]$?

Question

It satisfies every axiom to become a subset however I don't know if it contains the $0$ element or not. The set of $\mathbb{R}[x^3]$ is $1+x^3+x^6...$ so i don't see how it can contain the 0 element that is needed for it to be a subring.

Answer 1

Yes $0\in \Bbb R[x^3]$ just as it is in $\Bbb R[x]$. You get to multiply any finite number of $1,x^3,x^6,\ldots $ by reals and add them up to get an element of $\Bbb R[x^3]$. Otherwise $0$ would not be in $\Bbb R[x]$ either.

It really doesn't matter for the subset relation. $\Bbb R[x^3]\setminus\{0\}$ is a fine subset of $\Bbb R[x]$. It isn't a subring any more because $0$ is missing, but it is a subset.