Differentially trivial left Noetherian rings (Q2761017)

Let \(R\) be an associative ring with identity. An additive mapping \(D\colon R\to R\) is called a derivation of \(R\) if \(D(xy)=D(x)y+xD(y)\) for all \(x,y\in R\). A ring \(R\) is said to be differentially trivial if it has no non-zero derivations. A Noetherian ring \(R\) is differentially trivial if and only if it is of one of the following types: (i) \(R\) is a differentially trivial Noetherian domain (i.e. \(R\) is algebraic over its prime subring if \(\text{char}(R)=0\), and \(R=\{a^p\mid a\in R\}\) if \(\text{char}(R)=p\)); (ii) \(R\) is a subdirect product of finitely many differentially trivial Noetherian domains of characteristic \(0\); (iii) \(R\cong\bigoplus_{i=1}^n\mathbb{Z}_{p_i^{k_i}}\); (iv) \(R=F\oplus S\) is a ring direct sum, where \(S\) is a ring of type (ii) and \(F\) is a ring of type (iii).