Finitely generated, non-Artinian monolithic modules.
The paper under review provides a brief survey of the study of Noetherian rings over which the injective hull of every simple module is locally Artinian. Recall that a module is called locally Artinian if each of its finitely generated submodules is Artinian. For any ring \(R\), the condition that the injective hulls of simple right \(R\)-modules are locally Artinian is obviously equivalent to the condition that all finitely generated essential extensions of simple right \(R\)-modules are Artinian. By Matlis' theorem it follows clearly that every commutative Noetherian ring satisfies the property that injective hulls of simple right \(R\)-modules are locally Artinian. A. V. Jategaonkar and J. E. Roseblade showed that if \(G\) is a polycyclic-by-finite group then injective hulls of simple modules over the group ring \(R[G]\) are locally Artinian whenever \(R\) is a ring of characteristic zero, or \(R\) is a field that is algebraic over a finite field. Dahlberg showed that injective hulls of simple modules over \(U(\mathfrak{sl}_2)\) are locally Artinian. Jategaonkar showed that for a fully bounded Noetherian ring \(R\), the injective hull of every simple \(R\)-module is locally Artinian, and he used this fact to show that Jacobson's conjecture holds for a fully bounded Noetherian ring. A module \(M\) is called `monolithic' if the intersection of all nonzero submodules of \(M\) is nonzero (in other words, \(M\) has a unique minimal submodule). Note that for a Noetherian ring \(R\), the property that the injective hull of every simple \(R\)-module is locally Artinian is equivalent to the condition that every finitely generated monolithic \(R\)-module is Artinian. The author provides a general construction for finitely generated, non-Artinian, monolithic modules.NEWLINENEWLINE Georgia Benkart and Tom Roby introduced a unital associative algebra \(A=A(\alpha,\beta,\gamma)\) over a field \(K\) with generators \(u\), \(d\) and defining relations \(d^2u=\alpha dud+\beta ud^2+\gamma d\) and \(du^2=\alpha udu+\beta u^2d+\gamma u\) where \(\alpha\), \(\beta\), \(\gamma\) are fixed but arbitrary elements of \(K\). Benkart and Roby called these algebras `down-up algebras'. Since \(U(\mathfrak{sl}_2)\cong A(2,-1,-2)\), it motivated Patrick F. Smith to ask which Noetherian down-up algebras satisfy the property that finitely generated monolithic modules over them are Artinian. The author shows that if \(A(\alpha,\beta,\gamma)\) is a Noetherian down-up algebra over a field \(K\) of characteristic zero, then any finitely generated monolithic \(A(\alpha,\beta,\gamma)\)-module is Artinian if and only if the roots of the polynomial \(X^2-\alpha X-\beta\) are roots of unity.NEWLINENEWLINE This paper is very well-written and it will be useful for anyone interested in this area of research.NEWLINENEWLINEFor the entire collection see [Zbl 1232.16001].
scientific article; zbMATH DE number 6052239