Locally Artinian supplements related to Krasner hypermodules

Abstract

We introduce and investigate two new classes of Krasner left \(R\)-hypermodules: locally Artinian supplemented hypermodules and RLA\(-\)local hypermodules. A Krasner left \(R\)-hypermodule \(H\) is called locally Artinian if every finitely generated subhypermodule of \(H\) is Artinian, and it is called RLA\(-\)local if it is local and Rad\((H)\) is a locally Artinian subhypermodule of \(H\). A hypermodule \(H\) is called locally Artinian supplemented if every subhypermodule \(U\) of \(H\) has a locally Artinian supplement \(V\) in \(H\), meaning that
\(V\) is a supplement of \(U\) in \(H\) such that \(U \cap V\) is locally Artinian. We investigate fundamental closure properties of these classes, including closure under factor hypermodules and finite direct sums. We show that every RLA\(-\)local hypermodule is amply locally Artinian supplemented, and we characterize locally Artinian supplemented hypermodules in terms of their radicals. We further prove that for normal \(\pi\)-projective hypermodules, locally Artinian supplemented and amply locally Artinian supplemented conditions are equivalent. Several new results distinguishing these classes from ss-supplemented hypermodules and Artinian hypermodules are also investigated. Structural results concerning essential subhypermodules are established, and the relationship between essentiality and locally Artinian supplements is investigated, thereby situating the present study within the broader context of smallness and essentiality in Krasner hypermodules.