For a field, the ring-matrix operations, invertibility and similarity agree exactly with the established field-matrix interface

Given: A field $F$ regarded as a commutative ring.

A ring-valued $m\times n$ matrix is a function $m\times n\to F$ (Finite rectangular matrices over a commutative ring, their entries, rows and columns).

Ring-matrix operations, products, identities and transpose use the entrywise finite-sum formulas (Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose).

A field-valued matrix is a function $m\times n\to F$ (The vector space $M_{m \times n}(F) := F^{\,m \times n}$ of $m$ by $n$ matrices over a field, with entrywise operations).

Field-matrix products and identities use the same formulas (Rectangular matrix multiplication and the identity matrix $I_n$, including zero-sized shapes).

Field-matrix transpose interchanges the same entries (The transpose $A^{\mathsf T}$ of a matrix).

Every field is a commutative ring with the same operations and identities (Every field is a commutative ring with $1 \ne 0$; it is an integral domain, and it is a commutative division ring).

Ring-matrix invertibility and similarity are defined by $AB=BA=I$ and $B=P^{-1}AP$ (Invertible square matrices and similarity over a commutative ring).

Field-matrix invertibility uses the same two-sided inverse equations (Invertible matrices and the general linear group $\operatorname{GL}_n(F)$).

Field-matrix similarity uses the same conjugation equation (Similar matrices: $B=P^{-1}AP$ for an invertible $P$).