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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.

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  • $\begingroup$ Presumably you are considering rings with $1.\,$ If not, you should state that. $\endgroup$ – Bill Dubuque Jan 29 '15 at 15:24
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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.

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    $\begingroup$ Note that this nice description of the ideal requires the ring to be commutative and unital. Without these assumptions, the description get rather messy (but these seem like fair assumptions given the formulation of the question). $\endgroup$ – Tobias Kildetoft Jan 29 '15 at 9:54
  • $\begingroup$ @TobiasKildetoft Oooh, I guess you're right. Actually, after so much algebraic geometry and commutative algebra, I just forgot that there were rings which are non-commutative... $\endgroup$ – Amitai Yuval Jan 29 '15 at 10:01
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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

$$ \begin{array}{} && I \,\supset \{ r,\ s,\,\ldots\, \}\\ \iff && I \,\supset\, r R,\, s R,\, \ldots\\ \iff && I \,\supset\, r R + s R + \ldots \end{array}$$

But the latter set is already an ideal, so necessarily the smallest such ideal.

Remark $\ $ For principal ideals, where "contains" = "divides", the above is essentially the universal property of the gcd, namely

$$ i\mid r,s,\ldots\iff i\mid \gcd(r,s,\ldots)$$

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