Determinants of Matrices over Commutative Rings

Let \(R\) be a commutative ring. Fix an integer \(n \geq 1\). Let \(\mathbf{A} = (a_{ij})\) be the \(n \times n\) square matrix

$$\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{pmatrix}$$

with entry \(a_{ij}\in R\) and \(1 \leq i,j \leq n\) is at the intersection of the \(i^\mathrm{th}\) row and \(j^\mathrm{th}\) column. We will define a function \(\det(\mathbf{A}) =\det (a_{ij})\) of this matrix \(\mathbf{A}\), which will also be denoted by

$$\begin{vmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{n1} & a_{n2} & \cdots & a_{nn} \end{vmatrix}$$

We will also state and prove the various familiar properties of the determinant in this setting.
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