Suppose $a(x)$ and $b(x)$ are two non-zero polynomials in the polynomial field $F[x]$ have a gcd $d(x)$ can be expressed as a "linear combination": $$ d(x) = r(x)a(x) + s(x)b(x) $$ where $r(x)$ and $s(x)$ are in $F[x]$
Now if $J$ is the set of all the linear combinations:
$$ u(x)a(x) + v(x)b(x) $$
as $u(x)$ and $v(x)$ range over $F[x]$ then $J$ is an ideal of $F[x]$.
I don't understand this last statement "$J$ is an ideal of $F[x]$"
According to the definition of ideal:
"A nonempty subset $B$ of a ring $A$ is called an ideal of $A$ if $B$ is closed with respect to addition and negatives and $B$ absorbs products in $A$."
Now for the above problem how does this definition of ideal reasserts the idea that $J$ is an ideal?
Can anyone kindly help me find the answer?