Deleted-row-and-column minors, cofactors, the cofactor matrix and the adjugate over a commutative ring

Let $R$ be a commutative ring, let $n\ge 1$, and let $A=(a_{ij})\in M_n(R)$, with rows and columns indexed by $0,\dots ,n-1$ (Finite rectangular matrices over a commutative ring, their entries, rows and columns).

For $0\le i,j<n$, the deleted-row-and-column matrix $A^{(i,j)}$ is obtained by deleting row $i$ and column $j$. The minor and cofactor at $(i,j)$ are

$M_{ij}(A):=\det (A^{(i,j)}),C_{ij}(A):={\epsilon }_{ij}{M}_{ij}(A),$

where the sign is taken in $R$ by parity: ${\epsilon }_{ij}:=1_R$ when $i+j$ is even and ${\epsilon }_{ij}:=-1_R$ when $i+j$ is odd. Writing this as $(-1{)}^{i+j}$ is the usual abbreviation; it is the ring element $-1_R$ raised to a natural-number power in $R$, not a real power. When $n=1$, $A^{(0,0)}$ is the unique $0\times 0$ matrix and its determinant is defined locally to be $1$. For $n>1$, the determinant is the published matrix determinant (For $n\ge 1$, the determinant over a commutative ring by the Leibniz formula, and $\mathrm{\mid }\det A\mathrm{\mid }$ for a real matrix).

The cofactor matrix is $\mathrm{cof}(A):=(C_{ij}(A))$, and the adjugate is

$\mathrm{adj}(A):=\mathrm{cof}(A{)}^{\mathsf{T}},$

using the published transpose operation (Entrywise ring-matrix operations, rectangular matrix products, identity matrices and transpose).