en تخصصي Special پژوهشي Research paper <div style="text-align: justify;">Let $G$ be a finitely generated abelian group and $M$ be a $G$-graded $A$-module. In general, $G$-associated prime ideals to $M$ may not exist. In this paper, we introduce the concept of $G$-attached prime ideals to $M$ as a generalization of $G$-associated prime ideals which gives a connection between certain $G$-prime ideals and $G$-graded modules over a (not necessarily $G$-graded Noetherian) ring. We prove that the $G$-attached prime ideals exist for every nonzero $G$-graded module and this generalization is proper. We transfer many results of $G$-associated prime ideals to $G$-attached prime ideals and give some applications of it.</div> $G$-graded module, $G$-associated prime ideal, $G$-attached prime ideal, Weak $G$-attached prime ideal. 59 74 http://ijmsi.ir/browse.php?a_code=A-10-3880-1&slc_lang=en&sid=1 A. U. Ansari ajimmatau@gmail.com 10031947532846009578 10031947532846009578 Yes Department of Mathematics, University of Allahabad, Prayagraj, India B. K. Sharma brajeshsharma72@gmail.com 10031947532846009579 10031947532846009579 No Department of Mathematics, University of Allahabad, Prayagraj, India Sh. D. Kumar sdt@mnnit.ac.in 10031947532846009580 10031947532846009580 No Department of Mathematics, Motilal Nehru National Institute of Technology, Prayagraj, India S. Behara srinivasbehara45@gmail.com 10031947532846009581 10031947532846009581 No Government Polytechnic, Gannavaram, Krishna District, Andhra Pradesh, India