Semidistributive Laurent Series Rings

A module $$M$$ is said to be distributive (resp., uniserial) if the submodule lattice of a chain) Any uniserial but ring integers non-uniserial as $$\mathbb{Z}$$ -module. Direct sums (resp. modules are called semidistributive serial) modules. If $$A$$ with automorphism $$\varphi$$ , then we denote by $$A((x,\varphi))$$ skew Laurent series coefficient in which addition naturally defined and multiplication regard relation $$x^{n}a=\varphi^{n}(a)x^{n}$$ (for all elements $$a\in A$$ ). For $$\varphi=1_{A}$$ obtain ordinary $$A((x))$$ . It known that right semilocal only finite direct product Artinian rings $$A_{i}$$ $$\varphi(A_{i})=A_{i}$$ for $$i$$ In [11], it proved serial ring. both cases, The main result given paper Theorem 1.4, where prove following assertions. 1. ring, 2. Assume $$\varphi(e)=e$$ every local idempotent $$e\in Then this case, 3. We note there exists Artinuian not serial. Indeed, let $$F$$ field $$5$$ -dimensional -algebra generated $$3\times 3$$ matrices form $$\left(\begin{matrix}f_{11}&f_{12}&f_{13}\\ 0&f_{22}&0\\ 0&0&f_{33}\end{matrix}\right)$$ $$f_{ij}\in F$$ semidistributive, left serial, Therefore, also formal power $$A[[x]]$$ Artinian.