Square Matrix with Duplicate Rows has Zero Determinant/Proof 2

Theorem
If two rows of a square matrix over a commutative ring $\left({R, +, \circ}\right)$ are the same, then its determinant is zero.

Proof
Suppose that $\forall x \in R: x + x = 0 \implies x = 0$.

From Determinant with Rows Transposed, if you swap over two rows of a square matrix, the sign of its determinant changes.

If you swap over two identical rows of a matrix, then the sign of its determinant changes from $D$, say, to $-D$.

But the matrix is the same.

So $D = -D$ and so $D = 0$.