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 $-c$. A row scaling is reversible exactly when its scalar is a unit. These reversible transformations are the ring-level elementary row equivalences. When $R$ is a field, they 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. A scaling by a nonunit, including possibly $0$, is still a row transformation but is not called a row equivalence.