Artinian modules over groups of rank \(2\) (Q5954028)

The author studies some types of Artinian modules over the ring \(FG\), where \(F\) is a field and \(G\) is a free Abelian group of rank \(2\) (the problem of description of all Artinian modules over the ring \(\mathbb{Z} G\) is ``wild''). In some cases such a study can be reduced to the description of Artinian hypercentral modules over the ring \(FG\). The main result of the paper is a description of the \(FG\)-injective hull \(E\) of a monolithic Artinian module \(A\) over the ring \(FG\) where \(G=\langle g_1\rangle\times\langle g_2\rangle\) is the free Abelian group of rank \(2\): if \(A\) is \(FG\)-hypercentral, then \(E=P_1\oplus\cdots\oplus P_n\oplus\cdots\) where \(P_n=\bigoplus_{k\in\mathbb{N}}a_{n,k}F\) is a Prüfer \((g_1-1)\)-module, i.e. \(a_{n,1}(g_1-1)=0\), \(a_{n,k+1}(g_1-1)=a_{n,k}\) and \(P_1(g_2-1)=0\), \(a_{n+1,k}(g_2-1)=a_{n,k}\), \(n,k\in\mathbb{N}\). Earlier, \textit{L. A. Kurdachenko} [Ric. Mat. 44, No. 2, 303-335 (1995; Zbl 0919.20016)] studied the injective hull of Artinian modules over the ring \(\mathbb{Z} G\), where \(G\) is an Abelian group of finite rank.