LEC ALG3 4

Defined polynomial and power series rings. Noted that if $R$ is a division ring, so is $R((x))$.

[!Example] Ideals in $\mathbb{Z}[x]$
Let $\phi :\mathbb{Z}\to \mathbb{Z}[x]$ be the natural injection. If $I$ is an ideal in $\mathbb{Z}[x]$, ${\phi }^{-1}(I)$ must be an ideal in $\mathbb{Z}$. Thus, $n\mathbb{Z}\subseteq I$ for some $n$.

[!Exercise] $R^{op}$
$R^{op}$ is $R$ as a set. Denote the elements of $R^{op}$ by $a^{op}$ where $a\in R$. Define multiplication by $a^{op}{b}^{op}=(ba{)}^{op}$. Check:

1. $R$ is a division ring $⟺$ $R^{op}$ is
2. $(R^{op}{)}^{op}\cong R$.

Properties of ideals

Let $R$ be a commutative ring with $1$. Let $I,J$ be ideals of $R$.

1. $I+J=\{i+j|i\in I,j\in J\}$. This is a subring of $R$.
2. $IJ=\{i_1{j}_1+\cdots +i_n{j}_n|i_k\in I,j_k\in J,n\ge 1\}$.

$IJ\subseteq I\cap J$.

Generators of ideals

Let $I$ be an ideal. $X=\{a_i|a_i\in I\}\subseteq I$ is said to generate $I$ if every element of $I$ can be written as a finite sum $r_{i_1}{a}_{i_1}+\cdots +r_{i_k}{a}_{i_k}$, $r_i\in R$.

If $I=(a_1,\dots ,a_n)$ and $J=(b_1,\dots ,b_k)$, then $I+J=(a_1,\dots ,a_n,b_1,\dots ,b_k)$. $IJ=(a_i{b}_j|1\le i\le n,1\le j\le k)$.

3. $I(J_1+\cdots +J_k)=I{J}_1+\cdots +I{J}_k$.
4. $(IJ)K=I(JK)$.
5. $I:J=\{r\in R|rJ\subseteq I\}$.

[!Exercise]
$I=(x^2,xy)$, $J=(y)$. What is $I:J$?

[!Example]
Let $R=\mathbb{C}[x,y,z]$ or $\mathbb{R}[x,y,z]$ or $\mathbb{Q}[z,y,z]$. $I=(x^3-yz,y^2-xz,z^2-x^{2}y)$, $J=(x,y)$. Show that $I\cap J=(x^3-yz,y^2-xz)$.

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Prime and maximal ideals

[!Definition]
An ideal $P\subset R$ is prime if for any two ideals $I,J\in R$,

$$IJ\subseteq P⟹I\subseteq P\text{ or }J\subseteq P.$$

[!Lemma]
Let $P$ be an ideal in $R$. If for all $a,b\in R$, $ab\in P⟹a\in P\text{ or }b\in P$, then $P$ is a prime ideal.

[!corollary]
Let $R$ be a commutative ring with $1$. Then $P$ is a prime ideal $⟺$ $R/P$ is an integral domain.