Let \(R\) be a ring and \(S\) be a subset of \(R\text{.}\) The left ideal generated by \(S\) (resp., right ideal generated by \(S\) , ideal generated by \(S\)) is denoted by \((S)\) and it is defined as follows.
\begin{align*}
\text{Left ideal generated by }S:\quad\amp \left\{\sum_{i=1}^n x_is_i:x_i\in
R, s_i\in S\text{ and }n\in\N\right\};\\
\text{Right ideal generated by }S:\quad\amp \left\{\sum_{i=1}^n s_iy_i:y_i\in
R, s_i\in S\text{ and }n\in\N\right\};\\
\text{Ideal generated by }S:\quad\amp \left\{\sum_{i=1}^n x_is_iy_i:x_i,y_i\in
R, s_i\in S\text{ and }n\in\N\right\}.
\end{align*}
Lemma1.7.2.
Let \(\mathfrak{a}\) and \(\mathfrak{b}\) be left ideals (resp., right ideals, two-sided ideals) in a ring \(R\text{.}\) Then following are left ideals (resp., right ideals, two-sided ideals):
[Smallest left ideal containing both \(\mfa\) and \(\mfb\)].
\(\mathfrak{a}+\mathfrak{b}=\{a+b:a\in\mathfrak{a}\text{
and }b\in\mathfrak{b}\}\text{.}\) More generally if \(\mfa_i\) are left ideals of \(R\) then
\begin{equation*}
\sum\mfa_i=\left\{\sum_ix_i:x_i\in\mfa_i\text{ and all but finitely many}\quad
x_i=0\right\}.
\end{equation*}
[Smallest left ideal contained in both \(\mfa\) and \(\mfb\)].
Left to the reader. We prove more general version of (2) below.
Remark1.7.3.
In general \(\mfa\mfb\subsetneq\mfa\cap\mfb\subsetneq\mfa+\mfb\text{.}\)
Lemma1.7.4.
Let \(I\) be a nonempty indexing set, and let \(\mathfrak{a}_i\) (\(i\in
I\)) be left ideals (resp., right ideals, two-sided ideals) of a ring \(R\text{.}\) Then \(\bigcap_i\mathfrak{a}_i\) is a left ideal (resp., a right ideal, two-sided ideal) of \(R\text{.}\)
We only show that the intersection of left ideals is a left ideal. Suppose that \(x,y\in\bigcap_i\mathfrak{a}_i\text{.}\) Then, since each \(\mathfrak{a}_i\) is a subgroup, \(x-y\in\mathfrak{a}_i\) for each \(i\text{.}\) Hence \(\bigcap_i\mathfrak{a}_i\) is a subgroup. For any \(r\in R\) and any \(x\in\bigcap_i\mathfrak{a}_i\text{,}\) we have \(rx\in\mathfrak{a}_i\) because every \(\mathfrak{a}_i\) is a left ideal. Thus, \(rx\in\bigcap_i\mathfrak{a}_i\) and we get the result.
Observation1.7.5.
In view of Lemma 1.7.4 a left ideal (resp., right ideal, two-sided ideal) generated by \(S\) is the intersection of all left ideals in \(R\) containing \(S\text{,}\) i.e.,
\begin{equation*}
(S)=\bigcap_{\substack{S\subseteq
I\\I\text{ a left ideal}}}I.
\end{equation*}
