We study the problem of Regularized Unconstrained Submodular Maximization (RegularizedUSM) as defined by Bodek and Feldman [BF22]. In this problem, you are given a non-monotone non-negative submodular function ∶ 2 → ℝ ≥0 and a linear function ∶ 2 → ℝ over the same ground set , and the objective is to output a set ⊆ approximately maximizing the sum ( ) + ( ). Specifically, an algorithm is said to provide an ( , )-approximation for RegularizedUSM if it outputs a set such that [ ( ) + ( )] ≥ max ⊆ [ ⋅ ( ) + ⋅ ( )]. We also study the setting where and are subject to a matroid constraint, which we refer to as Regularized Constrained Submodular Maximization (RegularizedCSM).For both RegularizedUSM and RegularizedCSM, we provide improved ( , )-approximation algorithms for the cases of non-positive , non-negative , and unconstrained . In particular, for the case of unconstrained , we are the first to provide nontrivial ( , )-approximations for RegularizedCSM, and the we obtain for RegularizedUSM is superior to that of [BF22] for all ∈ (0, 1).In addition to approximation algorithms, we provide improved inapproximability results for all of the aforementioned cases. In particular, we show that the our algorithm obtains for RegularizedCSM with unconstrained is tight for ≥ +1 . We also show 0.478-inapproximability for maximizing a submodular function where and are subject to a cardinality constraint, improving the long-standing 0.491-inapproximability result due to Gharan and Vondrak [GV10].