1. ## prime

Show that the principal ideal $(x-1)$ in $\mathbb{Z}[x]$ is prime but not maximal.

To show it is prime consider $ab \in (x-1)$. Then we want to show that $a \in (x-1)$ or $b \in (x-1)$.

Now $(x-1) = \{c(x-1): c \in \mathbb{Z}[x] \}$. So $ab = c[x-1]$ for some polynomial $c \in \mathbb{Z}[x]$. This implies that either $a \in (x-1)$ or $b \in (x-1)$ since we can take the other to be the unit polynomial.

Suppose for contradiction that it was maximal. Then if $(x-1) \subseteq J \subseteq \mathbb{Z}[x]$, either $(x-1) = J$ or $J = \mathbb{Z}[x]$. Consider $(2x)$. Then it doesn't contain $x^2+1 = x(x+1)$. Thus $(x+1)$ is not maximal.

Is this correct?

2. Originally Posted by Sampras
Show that the principal ideal $(x-1)$ in $\mathbb{Z}[x]$ is prime but not maximal.

To show it is prime consider $ab \in (x-1)$. Then we want to show that $a \in (x-1)$ or $b \in (x-1)$.

Now $(x-1) = \{c(x-1): c \in \mathbb{Z}[x] \}$. So $ab = c[x-1]$ for some polynomial $c \in \mathbb{Z}[x]$. This implies that either $a \in (x-1)$ or $b \in (x-1)$ since we can take the other to be the unit polynomial.

Uuh?? What "other"? And what is "the unit polynomial"? Perhaps this is simpler: a polynomial is divisible by (x-1) iff 1 is one of its roots (this is the residue theorem for (long) division of polynomials), so $ab\in (x-1) \Longleftrightarrow ab(1)=0 \Longleftrightarrow a(1)=0 \;or \;b(1)=0...$

Suppose for contradiction that it was maximal. Then if $(x-1) \subseteq J \subseteq \mathbb{Z}[x]$, either $(x-1) = J$ or $J = \mathbb{Z}[x]$. Consider $(2x)$. Then it doesn't contain $x^2+1 = x(x+1)$. Thus $(x+1)$ is not maximal.

Is this correct?
Please do read now, coldly, what you wrote: can you understand it? Because if you can then you're not explaining it clearly at all, and from things like this marks in exams go away for vacation in Acapulco:

$(x-1)\subset (2,x-1)$

Tonio

3. What does the notation $(2,x-1)$ mean?

4. Originally Posted by Sampras
What does the notation $(2,x-1)$ mean?

Err...the ideal generated by x-1 and 2, also denoted by $2\mathbb{Z}[x] +(x-1)\mathbb{Z}[x]$

Tonio