Weak and Serre Lifting of Cyclic Modules and the Liftability of Auslander Duals

Let $Q$ be a local ring and $f$ a nonzerodivisor on $Q\mathrm{.}$ We investigate the weak and Serre lifting properties of cyclic modules over $R=Q\mathrm{/}(f)$ to $Q\mathrm{.}$ First, we establish characterization for the weak and Serre liftability of these modules and provide sufficient conditions under which their tensor products share these properties. We present an example that strengthens a counterexample of Nawaz and Levins and answers a question of Jorgensen. Finally, we also study the liftability of Auslander duals, obtaining a mild strengthening of a theorem of Ghosh and Samanta.