What is a linear combination, exactly? I'm used to the definition of linear combination used in linear algebra textbooks.
I'm reading the book Algebra by Artin and on page 357 he says:

If $R$ is the ring $\mathbb{Z}[x]$ of integer polynomials, the notation $(2,x)$ stands for the ideal of linear combinations of $2$ and $x$ with integer polynomial coefficients.

According to the definition above, the term $x\cdot x+2 = x^2+2$ is a linear combination of $x$ and $2$, which doesn't feel like linear to me.
Here is my question: What is the definition of a linear combination?
My confusion may arise because in a vector space there is no such thing as product of vectors.
 A: Let's look at a finite set of things, like $\{ x, y, z, w \}$.  Let's pretend they are just variables.  If we want to talk about a linear combination of these things, we first need to know what our "scalars" will be.
If we take our scalars to be all real numbers $\Bbb R$, then a linear combination of $\{x,y,z,w \}$ is a finite sum of these things with coefficients from the set of scalars, here $\Bbb R$.  So $\pi x + \sqrt{2} y$ is a linear combination since it is really $\pi x + \sqrt{2} y + 0z + 0w$.  As you can see, my coefficients, which are scalars, come from $\Bbb R$.
If we decided we wanted our scalars to be $\Bbb Z$, the integers, then the above would not be a linear combination of elements in $\{x,y,z, w\}$ since some scalars are not integers.  But an example of a linear combination with coefficients from $\Bbb Z$ is $19x + 5y - 3w$.
Now, we could choose our scalars to be wacky things, like the set of all $2 \times 2$ invertible matrices with real entries, for example.  Then a linear combination of the elements of $\{x,y,z,w\}$ over this set of scalars could be $\begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}x + \begin{bmatrix} 1 & 1 \\ 9 & 2 \end{bmatrix}z$.

Now, in your example, our coefficients, or scalars, happen to be integer polynomials.  For example, in your example, a scalar would be $3x^{2} + 5x$, since this is a polynomial with integer coefficients, and the polynomial itself is acting as our coefficient.  So, if we want to write a linear combination of $\{2, x \}$ with integer coefficients, we better write $oneThing * 2 + twoThing* x$ where $oneThing$ and $twoThing$ are coefficients (in this case, they better be polynomials with integer coefficients since these are our scalars).  This is why $x^{2} + 2 = x \cdot x + 2$ is a linear combination.  The $x$ has coefficient $x$, which is itself a polynomial with integer coefficients and thus a scalar.

If you now understand what a linear combination is above, the general way to write the definition of a linear combination of, say $\{ 2, x\}$ with scalars as polynomials with integer coefficients is to say "a linear combination is of the form $c_{1}2 + c_{2}x$ where $c_{1}$ and $c_{2}$ are scalars, i.e., polynomials with integer coefficients.
