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ON S-PURE SUBMODULES

Hassan Sabti Al-Rdeny Department of mathematic. College of Science. University of Baghdad. Baghdad. Iraq.
Bahar Hamad Al-Bahrani Department of mathematic. College of Science. University of Baghdad. Baghdad. Iraq.

Abstract

This paper explores various characterizations and properties of S-pure submodules, including their relationships with other types of submodules such as extended S-pure submodules and extended S-flat modules. We investigate conditions under which extended S-pure submodules are preserved under certain operations, such as taking direct sums and extensions. Additionally, we examine examples illustrating the significance of extended S-purity in different algebraic contexts, including its implications for projective modules and flat modules. We introduce some concepts class of modules relative to a semiradical property.  Let S semiradical property and M be a module. We say that N extended S-pure submodule if there is K submodule of M such that N∩K=0 and S(M/(NÅ K))=0.    In the first part, we define a submodule S2(M) by the equality S2(M)/(S(M) =S(M/S(M) ), for any module M. A module M is called S-Goldie module if S2(M)=M. Also, we introduce the concept extended S-pure submodule. We consider the relations between the second semiradical (S-Goldie) submodule and extended S-pure submodule. In the second part, we introduce the concept of extended S-flat module. M is called es-flat module if for any epimorphism f:N→M, kerf is extended S-pure submodule in N. Also, some basic properties are investigated.

Keywords

extended S-closed submodule, extended S-pure submodule, second semiradical of M, S-Goldie module, extended S-flat module,