Row swaps, arbitrary row scalings and row additions over a commutative ring, with reversible elementary cases distinguished

For a matrix over a commutative ring $R$, a row transformation is one of: interchanging two rows; multiplying one row by any scalar $c\in R$; or adding $c$ times one row to a distinct row.

A swap is reversible, as is row addition with reverse coefficient $-c$. Scaling by a unit is reversible, with inverse scaling by its reciprocal. For matrices with at least one column and a selected row, the scaling map on the full matrix space is reversible only when $c$ is a unit. For matrices with no columns, every scaling map is the identity, even when $c$ is a nonunit.

The designated ring-level elementary row equivalences are swaps, row additions, and scalings by units. When $R$ is a field, these are exactly the elementary row operations of Elementary row operations and row equivalence for finite matrices over a field, since every nonzero scalar is a unit. Scaling by a nonunit, including possibly $0$, remains a row transformation but is not designated an elementary row equivalence, including in the zero-column case.