[Paper Review] On A Class of Lifting Modules

Hatice Inankıl, Sait Halıcıoğlu+1 more|arXiv (Cornell University)|Dec 14, 2013

Rings, Modules, and Algebras2 references3 citations

TL;DR

This paper introduces principally $δ$-lifting and principally $δ$-semiperfect modules as generalizations of $δ$-lifting and $δ$-semiperfect modules, extending lifting module theory to cyclic submodules using $δ$-small submodules. The key contribution is proving that for a projective module $M$, $M$ is principally $δ$-semiperfect if and only if it is principally $δ$-lifting, and thus a ring $R$ is principally $δ$-semiperfect if and only if it is principally $δ$-lifting.

ABSTRACT

In this paper, we introduce principally $δ$-lifting modules which are analogous to $δ$-lifting modules and principally $δ$-semiperfect modules as a generalization of $δ$-semiperfect modules and investigate their properties.

Motivation & Objective

• To generalize $δ$-lifting and $δ$-semiperfect modules by focusing on cyclic submodules.
• To define and investigate principally $δ$-lifting modules using $δ$-small submodules.
• To characterize principally $δ$-semiperfect modules via projective $δ$-covers of factor modules.
• To establish structural equivalences between lifting and semiperfect properties in the principally $δ$-context.

Proposed method

• Define a cyclic submodule $mR$ to be $δ$-lifting if $M$ decomposes as $A \oplus B$ with $A \leq mR$ and $mR \cap B$ $δ$-small in $B$.
• Use $δ$-small submodules—defined via singular quotients—to generalize smallness in module decompositions.
• Introduce principally $δ$-semiperfect modules as those for which $M/mR$ admits a projective $δ$-cover for every $m \in M$.
• Apply Zorn’s Lemma to construct maximal submodules with singular quotients to characterize non-$δ$-small cyclic submodules.
• Use homological techniques, including lifting of homomorphisms and properties of projective modules, to prove decomposition results.
• Leverage the structure of Rad$_\delta(R)$ as a $δ$-small submodule to relate lifting and semiperfect conditions.

Experimental results

Research questions

• RQ1 When is a cyclic submodule $mR$ of a module $M$ said to have the $δ$-lifting property, and how does this generalize standard lifting modules?
• RQ2 What conditions ensure that a module $M$ is principally $δ$-semiperfect, and how does this relate to the existence of projective $δ$-covers of $M/mR$?
• RQ3 Under what conditions is a principally $δ$-lifting module also principally $δ$-semiperfect, and vice versa?
• RQ4 How do the properties of being principally $δ$-lifting or $δ$-semiperfect behave under direct sums and homomorphic images?
• RQ5 What is the precise relationship between a ring $R$ being principally $δ$-semiperfect and being principally $δ$-lifting?

Key findings

• A module $M$ is principally $δ$-lifting if and only if for every $m \in M$, $M$ decomposes as $A \oplus B$ with $A \leq mR$ and $mR \cap B$ $δ$-small in $B$.
• If $M_1$ is semisimple and $M_2$ is principally $δ$-lifting, and both are relatively projective, then $M = M_1 \oplus M_2$ is principally $δ$-lifting.
• Every principally $δ$-semiperfect module is principally $δ$-supplemented, and every homomorphic image or direct summand of such a module is also principally $δ$-semiperfect.
• For a projective module $M$, $M$ is principally $δ$-semiperfect if and only if it is principally $δ$-lifting.
• A ring $R$ is principally $δ$-semiperfect if and only if it is principally $δ$-lifting.
• A projective module $P$ is principally $δ$-lifting if and only if $P/\text{Rad}_\delta(P)$ is principally semisimple and every cyclic direct summand of the quotient lifts to a cyclic direct summand in $P$.