On J–Lifting Modules
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Abstract<p>Let R be a ring with identity and M is a unitary left R–module. M is called J–lifting module if for every submodule N of M, there exists a submodule K of N such that <inline-formula>
<tex-math><?CDATA ${\rm{M}} = {\rm{K}} \oplus \mathop {\rm{K}}\limits^\prime,\>\mathop {\rm{K}}\limits^\prime \subseteq {\rm{M}}$?></tex-math>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll">
<mrow>
<mi mathvariant="normal">M</mi>
<mo>=</mo>
<mi mathvariant="normal">K</mi>
<mo>⊕</mo>
<mover>
<mi mathvariant="normal">K</mi>
<mo>′</mo>
</mover>
<mo>,</mo>
<mi mathvariant="normal"> </mi>
<mover>
<mi mathvariant="normal">K</mi>
<mo>′</mo>
</mover>
<mo>⊆</mo>
<mi mathvariant="normal">M</mi>
</mrow>
</math>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012025_ieqn1.gif" xlink:type="simple"></inline-graphic>
</inline-formula> and <inline-formula>
<tex-math><?CDATA ${\rm{N}} \cap \mathop {\rm{K}}\limits^\prime { \ll _{\rm{J}}}\mathop {\rm{K}}\limits^\prime $?></tex-math>
<math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll">
<mrow>
<mi mathvariant="normal">N</mi>
<mo>∩</mo>
<mover>
<mi mathvariant="normal">K</mi>
<mo>′</mo>
</mover>
<msub>
<mo>≪</mo>
<mi mathvariant="normal">J</mi>
</msub>
<mover>
<mi mathvariant="normal">K</mi>
<mo>′</mo>
</mover>
</mrow>
</math>
<inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="JPCS_1530_1_012025_ieqn2.gif" xlink:type="simple"></inline-graphic>
</inline-formula>. The am of this paper is to introduce properties of J–lifting modules. Especially, we give characterizations of J–lifting modules.We introduce J–coessential submodule as a generalization of coessential submodule . Finally, we give some conditions under which the quotient and direct sum of J–lifting modules is J–lifting.</p>
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