The following is a problem from the Gallian book. I'm trying to understand what exactly this ideal is and how to verify that it is in fact an ideal.
"Let $R$ be a commutative ring with unity and let $a_1, a_2,..., a_n$ belong to $R$. Then $I = \langle a_1, a_2,..., a_n \rangle = \{r_1 a_1 + r_2 a_2 + ... + r_n a_n | r_i \in R\}$ is an ideal of $R$ called the ideal generated by $a_1, a_2,...,a_n$."
My interpretation of this is that we choose some set of $a_1, a_2,..., a_n$ from R to generate an ideal with. Then we choose every combination of $n$ elements from $R$, and call each set of them $r_1, r_2,...,r_i$ and apply them to our generator set. Is this accurate? If not I could use some educating and some guidance in verifying the ideal.
EDIT:
To clarify, my questions are: a) Is my interpretation of the generator correct? b) How to verify it is an ideal and is my proof sufficient?
Adding my approach to verifying that this is an ideal:
First prove that $a - b \in I$
$(r_1 a_1 + r_2 a_2 + ... + r_n a_n) - (s_1 a_1 + s_2 a_2 + ... + s_n a_n) \in I$
$(r_1 - s_1) a_1 + (r_2 - s_2) a_2 + ... + (r_n - s_n) a_n \in I$
Next prove that $ar$ and $ra$ are in $I$ whenever $a \in I$ and $r \in R$
$r(r_1 a_1 + r_2 a_2 + ... + r_n a_n) \in I$
$r r_1 a_1 + r r_2 a_2 + ... + r r_n a_n \in I$
And thus it's verified. Is this all logical?