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.
Best Answer
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 $64$.
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 $6$.