What does an ideal generated by a subset look like? I am acquainted with principal ideals relatively well but I am struggling to understand the more general definition of ideals generated by an arbitrary subset of a ring. I suppose my question comes down to, what does $RX$ really mean? Where $R$ a commutative ring and $X$ a subset of $R$. Examples would be helpful.
 A: Let $R$ be a ring, and $\{r_\alpha\}_{\alpha\in I}\subset R$. The ideal generated by $\{r_\alpha\}$ consists of all the finite sums$$\sum_{i=1}^na_ir_{\alpha_i},$$where the coefficients $a_i$ are just elements of $R$.
Take for example $\mathbb{Z}[x]$, the ring of polynomials in one variable with integer coefficients, and consider the ideal generated by $\{2,x\}$. It consists of all the sums $2a+bx,\quad a,b\in\mathbb{Z}[x]$. Note that this ideal cannot be generated by a single element.
A: Hint $ $ By definition, the ideal generated by a set is the smallest ideal containing the set. Now applying the closure propertes of an ideal we deduce
$$ \begin{array}{} & I \,\supseteq \,\{ a,\,\ b,\,\ldots\, \}\\[.2em]
 \iff & I \,\supseteq\, a R,\, b R,\, \ldots\\[.2em]
\smash[t]{\overset{\rm\color{#c00}U\!}\iff} & I \,\supseteq\, a R + b R + \ldots\\[.2em]
 &\ \ =\{ ar_1\! + br_2+ \ldots\, :\, a\in A, b\in B,\ r_i\in R\}
\end{array}\qquad$$
But it's easily checked that  latter set is already an ideal, so necessarily the smallest such ideal.
Remark $\ $ The final equivalence  uses $\,\rm\color{#c00}U = $ universal property of the ideal sum, which specializes to the gcd universal property in principal ideal domains, since there "contains = divides", i.e.
$$\begin{align} I\supseteq A,B,\ldots\!\! &\!\! \overset{\rm\color{#c00}U\!\!}\iff I\,\supseteq\ A+B+\cdots\\[.3em]
\leadsto\ \ i\mid a,b,\ldots\!\! &\iff i\mid \gcd(a,b,\ldots)
\end{align}$$
when specialized to principal ideals$\, A = (a),\ B = (b)\,$ in a PID.
