Last time we introduced the notion of an ideal in a ring. Stating it again

Definition: Ideal
An ideal in \(R\) is a subset \(I \subseteq R\) such that
  1. \(I\) is a subgroup of the abelian group with addition \((R, +)\). So the ideal must have \(0 \in I\), closed under addition, additive inverses are in \(I\) as well.
  2. If \(a \in I, r \in R\), then \(ra, ar \in I\) so it is also closed under multiplication.


We said that usually we refer to this definition as the “Two sided ideal” since other variants can exists (left and right ideals).

Proposition
If \(\{I_{\alpha}\}\) is a collection of ideals, then \(I = \bigcap_{\alpha} I_{\alpha}\) is an ideal.


As a consequence of this, we have the following defintion

Proposition
Given a subset \(S \subseteq R\) ring, define $$ \begin{align*} (S) = \bigcap_{\text{all ideas such that }S \subseteq I} I \end{align*} $$ \((S)\) is an Ideal in \(R\) generated by \(S\). It is in fact the smallest Ideal containing the subset \(S\).


It is the smallest, since by definition for any \(I \subseteq R\), \(S\) is contained in \(I\), so \((S)\) is also in \(I\). This is a nice formal definition but it’s hard to use. An explicit description of the Ideal is as follows

Proposition
If \(R\) is a ring with 1 and \(S \subseteq R\) then, $$ \begin{align*} (S) = \{0\} \cup \{a_1s_2b_1 + ... + a_ks_kb_k \ | \ k \geq 1, s_1,...s_k \in S, a_i,b_i \in R\} \end{align*} $$


Proof
Let \(T = \{a_1s_2b_1 + ... + a_ks_kb_k \ | \ k \geq 1, s_1,...s_k \in S, a_i,b_i \in R\}\). Then we need to show

  1. Step (1) is showing that \(T\) is an Ideal where \(S \subseteq T\)
  2. Step (2) is if any \(I \subseteq R\) is any idea such that \(S \subseteq I\), then \(T \subseteq I\)

(1) and (2) Together imply that \(T = (S)\).
[TODO: Write full proof]

Special Case: If \(R\) is a commutative ring, then

$$ \begin{align*} (S) = \{0\} \cup \{a_1s_2 + ... + a_ks_k \ | \ k \geq 1, s_1,...s_k \in S, a_i \in R\} \end{align*} $$



Principle Ideal

Another special case is the Principle Ideal.

Definition
A Principle Ideal \(I\) is an ideal such that \(I = (r)\) for a single \(r = R\) so $$ \begin{align*} (r) = \{a_1rb_1 + ... + a_krb_k \ | \ a_i,b_i \in R\} \end{align*} $$


Note that if \(R\) is a commutative ring with identity then

$$ \begin{align*} (r) &= \{a_1r + ... + a_kr \ | \ a_1,...a_k \in R\} \\ &= \{(a_1 + ... + a_k)r \ | \ a_1,...a_k \in R\} \\ &= \{ ar \ | \ a \in R \}. \end{align*} $$

Example: If \(K\) is a field. The only ideals are \(\{0\}\) and \(K\). \(\{0\}\) is generated by \((0)\) and \(K\) is generated by \((1)\).




References