Although what you are asking for is not true, there is a very interesting related fact that is true. Let $k$ be an algebraically closed field of any characteristic. The theory of matrix factorisations shows that every homogeneous polynomial in $R=k[X_1,\dots,X_n]$ factorises in a suitable matrix ring $\operatorname{Mat}_s(R)$ as a product of linear factors. This follows from Theorem 1.2 of Herzog, Ulrich and Backelin, "Linear maximal Cohen-Macaulay modules over strict complete intersections" (see also Backelin, Herzog and Sanders, "Matrix factorizations of homogeneous polynomials"). To illustrate this theorem, consider the polynomial $f=X_1^2+X_2^2+X_3^2+X_4^2$. Clearly, this does not factor into linear factors in $R$. But in $\operatorname{Mat}_4(R)$ we have $$f.I_4=\begin{pmatrix} X_1&-X_2&X_3&X_4\\X_2&X_1&-X_4&X_3\\-X_3&X_4&X_1&X_2\\-X_4&-X_3&-X_2&X_1\end{pmatrix}\begin{pmatrix} X_1&X_2&-X_3&-X_4\\-X_2&X_1&X_4&-X_3\\X_3&-X_4&X_1&-X_2\\X_4&X_3&X_2&X_1\end{pmatrix}.$$ where $I_4$ is a $4\times 4$ identity matrix.

The theorem is that if $f$ has degree $d$ then it factorises as a product of $d$ linear factors in a suitable size matrix ring over $R$. One consequence of this is that some power of $f$ is the determinant of a matrix whose entries are homogeneous linear polynomials (or zero).

Matrix factorisations come from the theory of maximal Cohen-Macaulay modules over hypersurfaces. I believe the idea originates in the work of David Eisenbud, but there were probably others involved too.

the definitionof the object/notation $R[X]$ when $R$ is a ring. It's not an assumption, it's a universally accepted notation/convention. Thus, when you mean something else by $R[X]$, for example its module structure only, it's best you specify this explicitly. As forthe obvious substitution, I (and probably others)didassume it exactly that way (see answer).But that's also the point - it's an assumption, not a convention, it's not canonical, nor it's implied by the definition of $R[X]$.$\endgroup$13more comments