Let $R$ be a ring (with $1$). Last night, I was trying to prove that $M_{n}(R)$ (the ring of $n \times n$ matrices over $R$) is a ring. As I have done in my previous linear algebra course (which was over $\mathbb{R}$ or $\mathbb{C}$), for the proof of painful associativity and distributivities of matrix ring, I looked for isomorphism between $M_{n}(R)$ and $\text{End}_{R}(R^{n})$ as the latter is eaiser to be seen as a ring. However, the construction of this map that I knew from linear algebra as follow was not necessarily an isomorphism unless $R$ is commutative:
- $F:\text{End}_{R}(R^{n}) \rightarrow M_{n}(R)$ defined by $\varphi \rightarrow (a_{ij})$ where $\varphi(e_{j}) = \sum_{i}a_{ij}e_{i}$ where $e_{j}$ is the unity of $j$th summand of $R^{n}$ when we consider multiplication or $\text{End}_{R}(R^{n})$ as $\circ$ where $(\varphi \circ \theta)(x) = \varphi(\theta(x))$ whenever $x \in R$.
What worked as an isomorphism is as follows:
- $F^{t}:\text{End}_{R}(R^{n}) \rightarrow M_{n}(R)$ defined by $\varphi \rightarrow F(\varphi)^{t}$ where we understand multiplication of $\text{End}_{R}(R^{n})$ as $(x)\varphi \theta = \theta(\varphi(x))$ whenever $x \in R$.
Just like this example, I wonder how many elementary mechanics break down when it comes to linear algebra over a ring or a commutative ring (with $1$).
(For an example regarding commutative ring $R$, see that invertible matrices in $M_{n}(R)$ are the ones whose determinants are units not just nonzero.)
Again, rings that I consider allow unity. I also want to require answers to be as "sharp" as possible, so if something does not work even when we assume that $R$ is integral domain, I am asking that to be specified. No require for rigorous proofs as this question is aimed for my own learning, but some nice examples are always appreciated (but not required). Thanks.
