We show that linear codes over $\mathbb {Z}_{p^{m}}$ satisfy two extended versions of Wei’s Duality Theorem with respect to generalized Hamming weights (GHW) and a natural extension of GHW. Our results use a different approach to obtaining Wei-type duality theorems by extending the well-known relation between GHW and column multiplicities for linear codes over finite fields. We also present several new bounds for the minimum Lee distance of linear codes over $\mathbb {Z}_{p^{m}}$ that arise from the Singleton-type bound with respect to GHW. Our bounds generalize and improve several existing minimum Lee distance bounds.