On Localized Baer-Type Rings via Idempotent Elements Under Structural Extensions

In this paper, we introduce and systematically investigate several classes of localized Baer-type rings associated with a fixed idempotent element e, namely e-Baer, e-quasi-Baer, e-$p.q.$-Baer, e-PP, and e-APP rings. We develop a unified structural framework relating these localized annihilator classes to weak localized zero-product conditions, including weak e-symmetric, weak e-reversible, weak e-reflexive, and weak e-semicommutative properties. We establish the principal implication relations among these classes and, under suitable hypotheses, obtain equivalence results connecting several localized and global conditions. Examples and counterexamples are provided to distinguish the classes and to demonstrate the failure of converse implications in general. We further study the behavior of localized annihilator conditions under various algebraic constructions and infinite operations. In particular, we establish permanence and transfer results for left e-APP rings under matrix, polynomial, Laurent polynomial, monoid, skew monoid, and skew polynomial extensions, as well as under Morita equivalence. For formal power series extensions, where the left $\mathrm{APP}$ property is not preserved in general, we obtain two sufficient conditions on the corner ring $eRe$ that ensure preservation of the left e-APP property. We also establish structural equivalences between left e-APP and left e-$p.q.$-Baer rings under suitable finiteness assumptions. These results provide a systematic connection between classical annihilator theory and its localized counterpart and extend the study of localized regularity and Baer-type properties in ring theory.