Let $K$ be a field of positive characteristic $p > 0$. We study the coactions of the Hopf algebra of the multiplicative group $H_{m}$ with underlying algebra $H = K \left[ X_{1},\cdots,X_{n} \right] / (X_{1}^{p^{s_{1}}},\cdots,X_{n}^{p^{s_{n}}})$, $n \ge 1$, $s_{1}\ge \cdots \ge s_{n} \ge 1$ on a $K$-algebra $A$. We give the rule for the set of additive endomorphism of $A$, that define a coaction of $H_{m}$ on $A$ commutative. For $s_{1} = \cdots = s_{n} = 1$, we obtain the explicit expression of such coactions in terms of $n$ derivations of $A$.