Question about a certain ideal

Question

I have a question regaridngs the following ideal:

$$I_a:=\lbrace af+Xg:f,g \in R[X]\rbrace\subseteq R[X]$$

If $a\in R^{\times}$ then $I_a$ is a principal ideal.

Now my question.

In my textbook the definition of a principal ideal was:

$$(a)=\lbrace ab:b\in R \rbrace$$

Suppose $I_a$ is a principal ideal, does that mean $I_a=\lbrace ab: b \in R[X]\rbrace$?

thanks for the answers :)

To prove that $I_a$ is principal you need to find $h(X) \in R[X]$ such that $I_a = (h(X))$, where $(h(X))$ denotes the set ${h(X)p(X) : p(X) \in R[X]}$. — azif00, Dec 16 '20 at 23:41

score 0 · Accepted Answer · answered Dec 17 '20 at 06:43

0

To remove this question from the unanswered queue and to provide a more comprehensive solution, principal ideals are those generated by a single element, so if $I$ is a principal ideal of a ring $R$, then $I = (r)$ for some $r \in R$ (there may be more than one possible generator). So it doesn't matter what ring $R$ you're working in, you can always prove that a given ideal $I$ is principal by showing that $I = (r) = \{sr|s\in R\}$ for some $r \in R$.

answered Dec 17 '20 at 06:43

Noah Solomon

1,922

Why is $\lbrace 2f+Xg:f,g\in \mathbb{Z}[X]\rbrace$ no principal ideal in $\mathbb{Z}[x]$ – CoffeeArabica Dec 17 '20 at 11:46
And if possible without arguing that $2$ is no unit – CoffeeArabica Dec 17 '20 at 11:47
@CoffeeArabica Check out this answer https://math.stackexchange.com/questions/36169/show-that-langle-2-x-rangle-is-not-a-principal-ideal-in-mathbb-z-x – Noah Solomon Dec 17 '20 at 16:21

Question about a certain ideal

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