sorry if this is a really obvious question but I am a first year and trying to get my head around the idea of "elementary basis transformations" to bases of finite dimensional vector spaces over a field (or more generally, finitely generated modules over a ring). From what I gather, this includes:
$\bullet$ Adding one basis element (or multiples of it) to another basis element. So if $\{b_1,b_2,b_3,\ldots b_n\}$ is a basis of $V$ over a field $F$, then $\{b_1+b_2, b_2,\ldots,b_n\}$ is a basis of $V$. As $\{b_1+b_2,b_2,b_1+b_2+b_3,\ldots,b_n\}$ etc.
$\bullet$ Multiplying a basis element by a non-zero element of $F$. So, $\{ab_1,b_2,\ldots, b_n\}$ is a basis of $V$ if $a\neq 0$.
Is this right? And if so, can we generalise this to free modules over a ring $R$ (any ring, even noncommutative?). I suppose we need to replace $a\neq 0$ to $a$ being a unit in the ring, but is that all we need to change? What about replacing $b_1$ by $b_1+2b_2$ in the basis. Does it matter if 2 is not a unit in the ring, e.g. $R=\mathbb{Z}$?