I understand the definition of an ideal, and in particular, that of a principal ideal. For example, consider the principle ideal $(2)$ of $\mathbb{Z}$. This is the set of all multiples of the integer $2$ – i.e. the even numbers. I understand this from the definition and from an intuitive point of view.
But when considering principal ideals of polynomial rings, I lose this intuition entirely. For example, consider the polynomial ring $\mathbb{F}_5[x]$ and the principal ideal $(x^4+x^2+x+1)$ of it. What does this consist of? In a sense, it must be multiples of $x^4+x^2+x+1$. Is this correct?
Any help would be appreciated!
Best Answer
Yes, in this context, it is right. I think the sense you're thinking of is called "set equality."
In any commutative ring $R$ with identity, for any $x\in R$, $(x)=xR$.
It's very easy to see. Firstly, $xR$ is an ideal containing $1\cdot x =x$, and any ideal containing $x$ must contain $xr$ for all $r\in R$, so $xR$ is the smallest ideal containing $x$.