Let $K$ be a commutative ring, and $A$ a possibly non-associative, non-commutative algebra, and non-unital algebra over $K$.
I'm curious about matrices and linear algebra over such general $K$-algebras, and I'm wondering whether there are some good references for this? Happy for references just for the associative but non-commutative case as well.
Weibel's Homological Algebra book discusses this a little bit in section 7.1, e.g. that if $A$ is associative but non-commutative, then the trace must take values in $A/[A,A]$ so that it is invariant under change of basis.