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Sia $R$ un anello di caratteristica diseguale. Si stabiliscono formule generali per gli endomorfismi di una differenziazione $F_a$ o $F_c$-iterativa di $R$ , con $c$ non zerodivisore di R. Tali formule sono note nel caso della caratteristica eguale.

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 $H=K[X_1,\mathrm{\cdots },X_n]/(X_1^{p^{s_1}},\mathrm{\cdots },X_n^{p^{s_n}})$, $n\ge 1$, $s_1\ge \mathrm{\cdots }\ge s_n\ge 1$ 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 $s_1=\mathrm{\cdots }=s_n=1$, we obtain the explicit expression of such coactions in terms of $n$ derivations of $A$.

Let $E$ be a finitely generated $R$-module, having finite projective dimension. We study the acyclicity of the approximation complex $𝒵(E)$ of $E$ in terms of certain Fitting conditions $F_k^{(i)}$ on the Fitting ideals of the $i$-th module of a projective resolution of $E$. We deduce some good properties of the symmetric algebra of $E$.

Let $(R,𝔪)$ be a standard graded $K$-algebra over a field $K$. Then $R$ can be written as $S/I$, where $I\subseteq {(x_1,...,x_n)}^2$ is a graded ideal of a polynomial ring $S=K[x_1,...,x_n]$. Assume that $n\ge 3$ and $I$ is a strongly stable monomial ideal. We study the symmetric algebra ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ of the first syzygy module ${\mathrm{Syz}}_1\left(𝔪\right)$ of $𝔪$. When the minimal generators of $I$ are all of degree 2, the dimension of ${\mathrm{Sym}}_R\left({\mathrm{Syz}}_1\left(𝔪\right)\right)$ is calculated and a lower bound for its depth is obtained. Under suitable conditions, this lower bound is reached.