
> 
> Consider the affine space given by four $2\times 2$ matrices, i.e., $\mathbb{A}^{16}\cong M(\mathbb{C})\_{2\times 2}^4$. Now, consider the algebraic set $V$ given by the vanishing of the relation $AB-CD=0$, where the matrices are as follows: $A=(a\_{ij}), B=(b\_{ij}), C=(c\_{ij})$ and $D=(d\_{ij})$.
> 
> 
> 


In other words, I have the following ring
$$R=\mathbb{C}[a\_{11},a\_{12},a\_{21},a\_{22}, b\_{11},\dotsc, d\_{21},d\_{22}]/I,$$ where $$I=(a\_{11}b\_{11}+a\_{12}b\_{21}−c\_{11}d\_{11}−c\_{12}d\_{21},\,a\_{11}b\_{12}+
a\_{12}b\_{22}−c\_{11}d\_{12}−c\_{12}d\_{22},\,a\_{21}b\_{11}+a\_{22}b\_{21}−c\_{21}d\_{11}−c\_{22}d\_{21},\,a\_{21}b\_{12}+a\_{22}b\_{22}−c\_{21}d\_{12}−c\_{22}d\_{22}).$$


The question is:



> 
> Prove that $\overline{a}\_{11}$ is a prime element in $R$.
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> 
> 


I don't know if there are any nice trick/strategy using commutative algebra (maybe using some change of coordinate or defining some norm of the ideal).
I'm thinking in terms of quiver representation. The above setup can be interpreted as:
$$\require{AMScd}\begin{CD}
\mathbb{C}^2 @>B>> \mathbb{C}^2\\
@VDVV @VVAV \\
\mathbb{C}^2 @>>C> \mathbb{C}^2.
\end{CD}$$


The ring $R$ is the coordinate ring of the representation space of the above quiver with relation, where the dimension vector is $(2,2,2,2).$


Any idea/suggestion will be apreciated.



Prove that $\overline{a}_{11}$ is a prime element in $R$
Now that I have thought about this further, I realize that you need much less than the Samuel Conjecture to solve this problem. By a dimension count, the ring $R$ is a complete intersection ring, hence Cohen-Macaulay. Thus, by the Unmixedness Theorem, to prove that the ideal $I=\langle a\_{1,1} \rangle$ is prime, it suffices to prove that the corresponding closed subscheme of $\text{Spec}(R)$ is irreducible and generically reduced. This can be proved using Bertini's Theorem and a homogeneity argument.


