Q:

Ideals Generated by an Element

I was wondering if anyone could give me a little explanation into ideals and principal ideals.  I know that for $a \in R$, the principal ideal generated by $a$ is the set 
$$\langle a \rangle = \{r_1as_1+r_2as_2+ \ldots +r_nas_n \ | \ r_i, s_i \in R \}$$
What I don't unerstand is how we choose the $r_i, s_i$. For example, if in $\mathbb{Z}$ we were looking for the principal ideal generated by $8$, that is $\langle 8 \rangle$, how would we write this set? The source of the confusion is the following problem:
In $\mathbb{Z}$, show that $8$ belongs to the ideal generated by $10$ and $16$ but $\langle 8 \rangle \neq \langle 10, 16 \rangle$.
When it says "ideal generated by $10$ and $16$", is this the same as the principal ideal generated by $10$ and $16$? And what does it mean to write $\langle 10, 16 \rangle$? How do we construct the principal ideal generated by two elements? The general idea is really confusing me. Thanks in advance for your help.

A:

What I don't unerstand is how we choose the $r_i$, $s_i$. For example, if in $\mathbb Z$ we were looking for the principal ideal generated by $8$, that is $\langle8\rangle$, how would we write this set?

In general the $r_i$ and $s_i$ may be any ring element, but in the case where the ambient ring $R$ is commutative – like $\mathbb Z$ – the definition of principal ideal may be considerably simplified.
$$
\langle a \rangle = \{ra\,:r\in R\}.
$$
I leave it as an exercise to you to prove that the definitions are equivalent in the commutative case. This means that $\langle8\rangle$ is simply the subset of $\mathbb Z$ consisting of multiples of $8$, such as $-16$, $0$, or $64$.

When it says "ideal generated by 10 and 16", is this the same as the principal ideal generated by 10 and 16? And what does it mean to write $\langle10,16\rangle$? How do we construct the principal ideal generated by two elements?

For one, be careful with the phrase "a principal ideal generated by two elements". An ideal is principal by definition if it is generated by a single element. In certain rings, such as $\mathbb Z$, every ideal is principal, but most other rings (like $\mathbb Z[x]$) have nonprincipal ideals.
More generally, when $R$ is commutative we may write an ideal generated by finitely many elements as
$$
\langle a_1,\dots,a_n\rangle = \{r_1a_1 + \cdots + r_na_n\,:\,r_i\in R\};
$$
in other words, the ideal consists of $R$-linear combinations of the $a_i$, in analogy with linear combinations of vectors in a vector space.
This means that $\langle10,16\rangle$ is the subset of $\mathbb Z$ consisting of $\mathbb Z$-linear combinations of $10$ and $16$, such as $10$, $26$, and $6$.

