What is the smallest ideal? I have hard time understanding what "the smallest ideal" means. I read on wikiepdia 
" If X is any subset of R, then the intersection of all left ideals of R containing X is a left ideal I of R containing X, and is clearly the smallest left ideal to do so."
Could someone please explain this with an example?
 A: This is a general principle:
Assume that a "foo" is a certain kind of set. Also assume that


*

*There exists at least one foo

*The intersection of arbitrarily many foos is also a foo


Then the intersection of all foos is a foo and, being a subset of any other foo, is rightfully called the smallest foo.
Often "foo" is someting like "bar containing $X$". In that case (and if the two conditins above hold), this smallest bar containg $X$ is often called "the bar generated (or spanned) by $X$". This way we can define the subgroup generated by a subset of a given group; the subspace spanned by a set of vectors of a vector space; or here the (left) ideal generated by a subset of a ring.
As a concrete example, conmsider the ring $\mathbb Z$ and the set $\{40,-30\}$. There are several ideals of $\mathbb Z$ that contain both $40$ and $-30$, namely $\mathbb Z,2\mathbb Z, 5\mathbb Z, 10\mathbb Z$ (whereas for example $15\mathbb Z$ contains only $-30$ and not $40$; and $8\mathbb Z$ contains $0$, but not $-30$). The intersection of all these is simply $10 \mathbb Z$. Hence the smalles ideal of $\mathbb Z$ containing both $40$ and $-30$.
A: For example, if $R=\mathbb{Z}$ and $X=\{2\}$, the smallest ideal that contains all the elements of $X$ is
$$2\mathbb{Z}=\{...,-4,-2,0,2,4,...\}$$
This is because, an ideal is itself a ring so it must contain all the multiples of the elements of $X$ and its possible sums (with multiples too).
A: Not really an example with ideals. But I think it's really helpful.
Consider 2 points $P$ and $Q$ in $\mathbb{R}^3$. let A be the subset of $\mathbb{R}^3$ wich is the intersection of all planes that contains $P$ and $Q$.
We can see directly that $A$ is the line connecting $P$ and $Q$ in $\mathbb{R}^{3}$. This is the smalles affine subspace wich contains $P$ and $Q$.
Note: an affine space in $\mathbb{R}^n$ is like a point, line, plane, ... (but these don't have to go trough the origin).
