What is the time complexity of multiplying two matrices over an arbitrary ring?

Question

I know that the time complexity of matrix multiplication over a field is well studied (multiplying two $n \times n$ matrices can be done in $n^\omega$ field operations, where $\omega$ is the matrix multiplication exponent). What about matrix multiplication over a commutative ring, such as a primary ideal domain? Is it the same?

Answer 1

Algebraic matrix multiplication algorithms can always be transformed into algorithms that compute bilinear decompositions $$XY = \sum_{k = 1}^{r(n)} \operatorname{tr} (F_k X) \operatorname{tr} (G_k Y) W_k$$ so that the number of operations increases at most by a constant multiple.

This works over every ring (and even semiring) and one can define the matrix multiplication exponent for every ring. All currently known matrix multiplication algorithms use integer constants and work over every ring, but this is not proven that this must hold in general. It is known that for fields the exponent of matrix multiplication depends only on the characteristic of the field (A. Schönhage, "Partial and total matrix multiplication", SICOMP 10(3)), and one can adapt this to prove that transcendental and integral extensions of rings do not change the matrix multiplication exponent.