I am having difficulty in understanding exactly the elements of the set $\mathbb{R}[x]/(x^3)$. I'll explain my thought process.
The Quotient Ring is the set of additive cosets, so we have that $$\mathbb{R}[x]/(x^3) = \{f+(x^3) : f\in\mathbb{R}[x]\}.$$ So we have the relation $$f-g\equiv 0 \mbox{ mod } x^3,$$ hence $x^3|f-g$.
Now, right now I understand this as a whole bunch of notation taken strictly from the definition. But what exactly are the elements of this quotient ring?
Best Answer
Let $p = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + \cdots + a_n x^n \in \mathbb{R}[x]$. Then the image of $p$ under the natural map $\mathbb{R}[x] \to \mathbb{R}[x]/\langle x^3 \rangle$ is $a_0 + a_1 x + a_2 x^2 + \langle x^3 \rangle$.
I.e., every element in $\mathbb{R}[x]/\langle x^3 \rangle$ is of the form $$(\text{polynomial of degree $2$}) + \langle x^3 \rangle$$