On Serge Lang's book Algebra, he wrote on page 748, after defining the module of differentials $\Omega^1_{A/R}$ of an $R$-algebra $A$ and higher differentials $\Omega^i_{A/R}=\wedge^i \Omega^1_{A/R}$, that there exists a unique sequence of $R$-homomorphisms $d_i:\Omega^i_{A/R}\rightarrow\Omega^{i+1}_{A/R}$ such that for $\omega \in\Omega^i_{A/R}$ and $\eta\in\Omega^j_{A/R}$ we have $d(\omega\wedge\eta)=d\omega\wedge\eta+(-1)^i\omega\wedge d\eta$. He then pointed out $d^2=0$ as a consequence of the definition.
The question is, as I saw elsewhere, people usually impose one more axiom, that $d^2r=0$ for $r\in R$, so that we can define $d$ inductively. And I think Lang is missing something here. So, is it unnecessary to require $d^2r=0$ here?