Classical 1-Absorbing Primary Submodules

Abstract

Over the years, prime submodules and their generalizations have played a pivotal role in commutative algebra, garnering considerable attention from numerous researchers and scholars in the field. This papers presents a generalization of 1-absorbing primary ideals, namely the classical 1-absorbing primary submodules. Let R be a commutative ring and M an R-module. A proper submodule K of M is called a classical 1-absorbing primary submodule of M, if xyz eta is an element of K for some eta is an element of M and nonunits x,y,z is an element of R, then xy eta is an element of K or zt eta is an element of K for some t >= 1. In addition to providing various characterizations of classical 1-absorbing primary submodules, we examine relationships between classical 1-absorbing primary submodules and 1-absorbing primary submodules. We also explore the properties of classical 1-absorbing primary submodules under homomorphism in factor modules, the localization modules and Cartesian product of modules. Finally, we investigate this class of submodules in amalgamated duplication of modules.