On derivations of finite index, chain conditions, dimensions and radicals
This is an appealing and interesting paper in which the author studies the relation between chain conditions for an R module M and M considered as an R module with derivation. Specifically, for a left R module M and derivation d of R, an additive mapping D of M is a d-derivation if for all \(r\in R\) and \(m\in M\), \(D(rm)=d(r)m+rD(m)\). This notion is equivalent to M being a left module for the Ore extension ring R[x;d], although this point of view is not used by the author. Define D to be of finite index if for some integer \(k>0\), and each R submodule N of M, the R module \(N+D(N)+...+D^{k-1}(N)\) is D invariant. Finally let \({}_ DM\) mean that M is considered as an R[x;d] module. The author assumes throughout that D is a d-derivation of R of finite index k, and proves that \({}_ RM\) is Noetherian, or Artinian, exactly when \({}_ DM\) is; the length, or uniform dimension, of \({}_ RM\) is at least that of \({}_ DM\) and is bounded by k times that of \({}_ DM\); and when either exists, the Krull (or Gabriel) dimension of \({}_ RM\) equals that of \({}_ DM\). The final result in the paper assumes in addition that d itself is of finite index acting on R and that the index is invertible in R. In this case, if \({}_ DM\) is a simple module then \({}_ RM\) is completely reducible of length at most k, and so, when \({}_ dR\) is a simple module it is a simple Artinian ring. The heart of the proofs is the middle section of the paper which is purely lattice theoretic in nature.
scientific article; zbMATH DE number 4017122