Definition:

A subset $I\vartriangleleft R$ of a ring is called an ideal if:

1. It is an additive subgroup of $R$ (it contains 0 and is closed under $\pm$)
2. $I$ is closed under multiplication by an arbitrary element of $R$.

$$ \forall x\in I, r\in R,\\ rx,xr\in I $$

1. It is a proper subset since $1_R\not\in I$
   1. If $1\in I$ then $\forall\ r \in R, r= r\cdot 1 \in I\implies R = I$

$R$ is not an ideal of itself, it is an “improper ideal”

Examples:

1. $f:\mathbb{Z}\rightarrow\mathbb{Z}$

   $a\mapsto a\pmod n$

   $\ker f =\{a\in\mathbb{Z}:a\equiv0\pmod n\}=\{...,-2n, -n,0,n,2n,...\}= n\Bbb{Z}$

   $f$ is an ideal

   What do ideals of $\mathbb{Z}$ look like?

   Any $I\vartriangleleft \mathbb{Z}$ is an additive subgroup of $(\mathbb{Z},+)$ and is cyclic: $m\mathbb{Z}$ for some integer $m$

2. Let $F$ be a field.

   $f:F[x]\rightarrow F$

   $f(p(x))=p(0)=a_0+a_1(0)+a_2(0)^2+\cdots+a_n(0)^n=a_0$

   Then $f$ is a ring homomoprhism.

   $\ker f =\{p(x)\in F[x]:p(x)=a_1x+a_2x^2+\cdots+a_nx^n\}$, the set where the leading coefficient is equal to 0.

   Given $\alpha\in F,$

   $f:F[x]\rightarrow F$

   $f(p(x))=p(\alpha)$

   If $\alpha=1,$

   $f(p(x))=\sum_{k=0}^na_k$, the sum of all coefficients.

   $\ker f=\{p(x)\in F[x]:p(\alpha)=0\}$

3. $f:\mathbb{Z}[i]\rightarrow\mathbb{Z}_5$

   $f(a+bi)=(a+2b)\pmod 5$

   Exercise: Verify $f$ is a ring homomorphism. Is there a homomorphism $g:\mathbb{Z}[i]\rightarrow\mathbb{Z}_7$? (more difficu

   $\ker f=\{a+bi:a,b\in\mathbb{Z}:a+2b\equiv0\pmod 5\}$ (equivalently $a\equiv3b\pmod 5)$

   For example, $4+3i\in\ker f$

4. Suppose $F$ is a field.

   What are the ideals of $F$?

   $\{0_F\}\vartriangleleft F$

   Let $I\vartriangleleft F$ be an ideal of a field.

   If $0\not=x\in I.$ Then, $x^{-1}\in I$.

   But then $x^{-1}x=1\in I$ which is a contradiction.

   Thus, $\{0_F\}$ is the only ideal of a field.

Corollary:

If $I\vartriangleleft R$, then $I$ contains no units.

Exercise:

If $F$ is a field, then the only ideal of $M_n(F) \text{ is } \{0\}$

For the rest of the lecture, $R$ is a commutative ring.

(some of this can be found in section 5.4 of the textbook)

Definition:

Let $S\subseteq R$ be a subset.

The ideal generated by $S$, denoted $\langle\ S\ \rangle \vartriangleleft R,$ is

$$ \boxed{\langle\ S\ \rangle = \left\{ \sum_{i=1}^m r_is_i:r_i\in R,s_i \in S\right\}} $$

Exercise: