GCD in multivariate polynomial ring and quotient ring 1. GCD in multivariate polynomial ring   


I would like to prove the following but couldn't figure out how to.
Let $d$ and $h_1, h_2, \cdots, h_k$ be multivariate polynomials with $l$ number of indeterminates, i.e. $d, h_1, h_2, \cdots, h_k \in \mathbb{C}[x_1, x_2, \cdots, x_l]$. If
$$
    \tag 1 a \cdot d = b_1 \cdot h_1 + b_2 \cdot h_2 + \cdots + b_k \cdot h_k 
$$holds for arbitrary $a, b_1, b_2, \cdots, b_k \in \mathbb{C}[x_1, x_2, \cdots, x_l]$, then $d$ is a (greatest) common divisor (GCD) of $h_1, h_2, \cdots, h_k$.
 I think I can prove that the converse is true, i.e. if $d$ is a GCD
of $h_1, h_2, \cdots, h_k$, then $(1)$ holds, but I have no clue how to proceed with the original claim. Does Hilbert's Nullstellensatz somehow come into play in proving the above claim?
 Also, if $(1)$ holds, can I say that the principal ideal generated by $d$ equals the ideal generated by $h_1, h_2, \cdots, h_k$, i.e. $&lt;d&gt; = &lt;h_1, h_2, \cdots, h_k&gt;$? Can this be true even if the multivariate polynomial ring is not a principal ideal domain?    

2. GCD in multivariate polynomial quotient ring  


Now I have an ideal $I$ in $\mathbb{C}[x_1, x_2, \cdots, x_l]$. Let $d$ and $h_1, h_2, \cdots, h_k$ be multivariate polynomials  in the quotient ring $R=\mathbb{C}[x_1, x_2, \cdots, x_l]/I$. If
$$
\tag 2 a \odot d = b_1 \odot h_1 + b_2 \odot h_2 + \cdots + b_k \odot h_k 
$$holds for arbitrary $a, b_1, b_2, \cdots, b_k \in R$
, where $\odot$ denotes the polynomial multiplication operation in the quotient ring $R$, can I call $d$ a (greatest) common divisor of $h_1, h_2, \cdots, h_k$ in $R$? How would I compute $d$ given $h_1, h_2, \cdots, h_k$ in this case?
The following is my initial thoughts. Since I know how to compute the Groebner basis $G=\{g_1, g_2, \cdots, g_m\}$ of $I$, I can lift $(2)$ to $\mathbb{C}[x_1, x_2, \cdots, x_l]$ and have 
$$
\tag 3 a \cdot d + r_1 = b_1 \cdot h_1 + b_2 \cdot h_2 + \cdots + b_k \cdot h_k + r_2 
$$where $r_1$ and $r_2$ are unique polynomials in $&lt;G&gt;$. Collecting the terms, I get
$$
\tag 4 a \cdot d = b_1 \cdot h_1 + b_2 \cdot h_2 + \cdots + b_k \cdot h_k + r_3
$$where $r_3 = r_2-r_1$. Since $r_3 \in &lt;G&gt;$, I can write the above into
$$
\tag 5 a \cdot d = b_1 \cdot h_1 + b_2 \cdot h_2 + \cdots + b_k \cdot h_k + c_1 \cdot g_1 + c_2 \cdot g_2 + \cdots + c_m \cdot g_m
$$ for some $c_i$.
Now $(5)$ looks awfully like $(1)$ so I want to say $d$ is a GCD of $\{h_1, h_2, \cdots, h_k, g_1, g_2, \cdots, g_m\}$.
Am I on the right track?
I have only taken linear algebra and no other algebra course but I have access to most textbooks. A gentle nudge in the right direction and pointers to relevant materials will be greatly appreciated.
First, $d$ is a common divisor of $h_1,\ldots,h_k$.

Second, if $s$ is a common divisor of $h_1,\ldots, h_k$, then $s$ divides $d$.
First, $d$ is a common divisor of $h_1,\ldots,h_k$.

Second, if $s$ is a common divisor of $h_1,\ldots, h_k$, then $s$ divides $d$.