More explicitly, if $M_{2 \times 2}(\mathbb{R})[x]$ denotes the ring of polynomials over the ring of 2x2 matrices with real coefficients (with indeterminate x a 2 by 2 matrix with real coefficients), how do i properly define multiplication? e.g. suppose $A_0,A_1,B_0,B_1 \in M_{2 \times 2}(\mathbb{R})$, and let $f[x] = A_0 + A_1x$ and $g[x] = B_0 + B_1x$ be elements of $M_{2 \times 2}(\mathbb{R})[x]$. Then I would *assume* the product $fg[x]$ would be

$ fg[x] = (A_0 + A_1x)(B_0 + B_1x) = A_0B_0 + A_0B_1x + A_1xB_0 + A_1xB_1x $

But then complications arise due to $M_{2 \times 2}(\mathbb{R})$ being non-commutative. As far as I know (I've only taken one course so far on abstract algebra), this ring is well-defined (in that a polynomial ring can have coefficients in *any* ring, not just commutative ones). I've checked google, wikipedia, etc and haven't found anything relevant to this topic. Is there any standard literature on this topic?

My plan was to eventually be able to investigate cases in which unique factorizations may hold (if any), or maybe polynomials having unique left or right inverses, etc.