IHJPAS. 36(2)2023 407 This work is licensed under a Creative Commons Attribution 4.0 International License Abstract The concept of the Extend Nearly Pseudo Quasi-2-Absorbing submodules was recently introduced by Omar A. Abdullah and Haibat K. Mohammadali in 2022, where he studies this concept and it is relationship to previous generalizationsm especially 2-Absorbing submodule and Quasi-2- Absorbing submodule, in addition to studying the most important Propositions, charactarizations and Examples. Now in this research, which is considered a continuation of the definition that was presented earlier, which is the Extend Nearly Pseudo Quasi-2-Absorbing submodules, we have completed the study of this concept in multiplication modules. And the relationship between the Extend Nearly Pseudo Quasi-2-Absorbing submodule and Extend Nearly Pseudo Quasi-2- Absorbing ideal. We also studied more result of Extend Nearly Pseudo Quasi-2-Absorbing submodule in multiplication module. In the end, we obtained new Propositions and distinguished results in studying this concept. Keywords: EXNPQ-2-Absorbing submodule, multiplication modules, non-singular modules, faithful module, projective module, good rings and local rings. 1. Introduction In recent years, many generalizations have appeared about the concept of the 2-Absorbing submodule such as (Pseudo Quasi-2-Absorbing, Nearly Quasi-2-Absorbing and Soc-QP2- Absorbing) submodules see [1, 2 and 3]. The concept of the Extend Nearly Pseudo Quasi-2- Absorbing submodules is one of the recent generalizations that were recently introduced by us, researchers, Omar and Haibat see [4]. Where we dealt with in the previous research basic properties with relationships. The present work is divided into three parts. Part one is preliminaries part, we present in this part of the work the necessary background needed later consisting of definitions, propositions and remarks (without proof) and in the second part we introduced and studied the concept of the Extend Nearly Pseudo Quasi-2-Absorbing submodule in multiplication module. Also we got a lot of important results like Propositions 3.2, 3.6 and 3.7. In the end we doi.org/10.30526/36.2.3045 Article history: Received 16 September 2022, Accepted 6 November 2022, Published in April 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Extend Nearly Pseudo Quasi-2-Absorbing submodules(II) Omar A. Abdullah Department of Mathematics College of Computer Science and Mathematics Tikrit University / Iraq. omer.a.abdullah35383@st.tu.edu.iq Haibat K. Mohammadali Department of Mathematics College of Computer Science and Mathematics Tikrit University / Iraq. H.mohammadali@tu.edu.iq https://creativecommons.org/licenses/by/4.0/ mailto:omer.a.abdullah35383@st.tu.edu.iq mailto:H.mohammadali@tu.edu.iq mailto:H.mohammadali@tu.edu.iq IHJPAS. 36(2)2023 408 presented more result of Extend Nearly Pseudo Quasi-2-Absorbing submodule in multiplication modules. See Propositions 4.1, 4.2 and 4.10. 2. Preliminaries The following list some fundamental definitions and notations that will be utilized in this paper. Definition 2.1[4]. A proper submodule 𝑉 of an Ζ¦-module Ρ  is said to be Extend Nearly Pseudo Quasi-2-Absorbing ( for short EXNPQ2AB ) submodule of Ρ  if whenever ɑɓ𝑐ӽ ∈ 𝑉, where Ι‘,Ι“, 𝑐 ∈ Ζ¦, Σ½ ∈ Ρ , implies that either ɑ𝑐ӽ ∈ 𝑉 + π‘ π‘œπ‘(Ρ )+𝐽(Ρ ) or ɓ𝑐ӽ ∈ 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ι‘Ι“Σ½ ∈ 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). And an ideal Ζ€ of a ring Ζ¦ is called EXNPQ2AB ideal of Ζ¦, if Ζ€ is an EXNPQ2AB Ζ¦- submodule of an Ζ¦-module Ζ¦. Definition 2.2[5]. An Ζ¦-module Ρ  is multiplicatiion, if every submodule 𝑉 of Ρ  is of the form 𝑉 = Ζ€Ρ  for some ideal Ζ€ of Ζ¦. Equivalently Ρ  is a multiplicatiion Ζ¦-module if every submodule 𝑉 of Ρ  of the form 𝑉 = [𝑉:Ζ¦ Ρ ]Ρ . Definition 2.3[6]. An Ζ¦-module Ρ  is faithful if π‘Žπ‘›π‘›Ζ¦(Ρ ) = (0), where π‘Žπ‘›π‘›Ζ¦(Ρ ) = {π‘Ÿ ∈ Ζ¦: π‘Ÿπ‘€ = (0)}. Definition 2.4[6]. An Ζ¦-module Ρ  is finitely generated if Ρ  = Ζ¦π‘₯1 + Ζ¦π‘₯2 + β‹― + Ζ¦π‘₯𝑛 for π‘₯1, π‘₯2,….., π‘₯𝑛 ∈ Ρ . Definition 2.5[7]. An Ζ¦-module Ρ  is called concellation module if Ζ€Ρ  = ƁѠ for any ideals Ζ€ and Ɓ of Ζ¦ implies that Ζ€ = Ɓ. Lemma 2.6[ 5, Coro. (2.14) (i)]. Let Ρ  be faithful multiplication Ζ¦-module, then π‘ π‘œπ‘(Ζ¦)Ρ  = π‘ π‘œπ‘(Ρ ). Lemma 2.7 [ 8, Coro. (2.14) (i)]. Let Ρ  be faithful multiplication Ζ¦-module, then 𝐽(Ζ¦)Ρ  = 𝐽(Ρ ). Definition 2.8[6]. An Ζ¦-module Ρ  is a projective if π‘“π‘œπ‘Ÿ π‘Žπ‘›π‘¦ Ζ¦-epimorphism 𝑓 from an Ζ¦-module Ρ  on to an Ζ¦- module Ρ Μ… and for any homomorphism 𝑔 from an Ζ¦-module Ρ ΜΏ to Ρ Μ…, there exists a homomorphism β„Ž from Ρ ΜΏ to Ρ  such that 𝑓 ∘ β„Ž = 𝑔. Lemma 2.9[ 6, Theo. (9.2.1) (g)]. For any projective Ζ¦-module Ρ , we have 𝐽(Ζ¦)Ρ  = 𝐽(Ρ ). Lemma 2.10[ 8, Prop. (3.24)]. For any projective Ζ¦-module Ρ , we have π‘ π‘œπ‘(Ζ¦)Ρ  = π‘ π‘œπ‘(Ρ ). Remark 2.11[6]. Ζ¦ is a good ring if 𝐽(Ζ¦)Ρ  = 𝐽(Ρ ). Definition 2.12[9]. Aring Ζ¦ is Artinian if Ζ¦ satisfies (DCC) is an ideals of Ζ¦, that is if {ƀ∝}βˆβˆˆβ‹€ is a family of ideals of Ζ¦ such that Ζ€1 βŠ‡ Ζ€2 βŠ‡ β‹― , then βˆƒΙ± ∈ Θ€ + such that ƀ𝑛 = Ζ€Ι± for any 𝑛 β‰₯ Ι±. Definition 2.13[10]. Aring Ζ¦ is said to be local ring Ζ¦ if Ζ¦ has a unique maximal ideal. IHJPAS. 36(2)2023 409 Lemma 2.14[ 6, Coro. (9.7.3) (b)]. If Ζ¦ is an Artinian ring, then Ζ¦ is a good ring. Lemma 2.15[ 11, Prop. (1.12)]. If Ρ  is an Ζ¦-module over local ring Ζ¦, then 𝐽(Ζ¦)Ρ  = 𝐽(Ρ ). Definition 2.16[12]. An Ζ¦-module Ρ  is π‘›π‘œπ‘›-π‘ π‘–π‘›π‘”π‘’π‘™π‘Žπ‘Ÿ if Θ€(Ρ ) = Ρ , where Θ€(Ρ ) = {π‘₯ ∈ Ρ : π‘₯Ζ€ = (0), for some essential ideal Ζ€ of R}. Lemma 2.17[ 12, Coro. (1.26)]. Let Ρ  be is a non-singular Ζ¦-modules, then π‘ π‘œπ‘(Ζ¦)Ρ  = π‘ π‘œπ‘(Ρ ). Lemma 2.18[ 13, Coro of Theo. (9)]. Let Ρ  be a finitely generated multiplication Ζ¦-module Ζ€ and Ɓ are ideals of Ζ¦. Then Ζ€Ρ  βŠ† ƁѠ if and only if Ζ€ βŠ† Ɓ + π‘Žπ‘›π‘›Ζ¦(Ρ ). Definition 2.19[14]. An Ζ¦-module Ρ  is called a 𝑍-regular if for each 𝑒 ∈ Ρ  π‘‘β„Žπ‘’π‘Ÿπ‘’ 𝑒π‘₯𝑖𝑠𝑑𝑠 𝑓 ∈ Ρ β€² = π»π‘œπ‘šΖ¦(Ρ , Ζ¦) such that 𝑒 = 𝑓(𝑒)𝑒. Definition 2.20[15]. An Ζ¦-module Ρ  is called weak cancellation if ƁѠ = Ζ€Ρ , implies that Ɓ + π‘Žπ‘›π‘›Ζ¦(Ρ ) = Ζ€ + π‘Žπ‘›π‘›Ζ¦(Ρ ) for Ɓ, Ζ€ are ideals in Ζ¦. Lemma 2.21[ 8, Prop. (3.25)]. Let Ρ  be a 𝑍-regular Ζ¦-module, then π‘ π‘œπ‘(Ρ ) = π‘ π‘œπ‘(Ζ¦)Ρ . Lemma 2.22[ 7, Prop. (3.9)]. If Ρ  is a multiplication Ζ¦-module, then Ρ  is finitely generated if and only if Ρ  is weak cancellation. Lemma 2.23[ 7, Prop. (3.1)]. If Ρ  is a multiplication Ζ¦-module, then Ρ  is concellation if and only if Ρ  is faithful finitely generated. Proposition 2.24[ 4, Prop. (3.4)]. A proper submodule 𝑉 of Ρ  is EXNPQ2AB submodule of Ρ  if and only if ɑɓ𝑐ℒ βŠ† 𝑉, for Ι‘, Ι“, 𝑐 ∈ Ζ¦ and β„’ is a submodule of Ρ , implies that either ɑ𝑐ℒ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or ɓ𝑐ℒ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ι‘Ι“β„’ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Proposition 2.25[ 4, Prop. (3.5)]. Let Ρ  be module and 𝑉 βŠ‚ Ρ . Then 𝑉 is EXNPQ2AB submodule of Ρ  if and only if for every submodule 𝐴 of Ρ  and for every ideals Ζ€1, Ζ€2, Ζ€3 of Ζ¦ such that Ζ€1Ζ€2Ζ€3𝐴 βŠ† 𝑉, implies that either Ζ€1Ζ€2𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Proposition 2.26[ 4, Coro. (3.7)]. Let Ρ  be an Ζ¦-module and 𝑉 βŠ‚ Ρ . Then 𝑉 is EXNPQ2AB submodule of Ρ  if and only if for each π‘Ÿ ∈ Ζ¦, π‘₯ ∈ Ρ  and every ideals Ζ€, 𝐽 of Ζ¦ with π‘ŸΖ€π½π‘₯ βŠ† 𝑉, implies that either π‘ŸΖ€π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Ÿπ½π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or ƀ𝐽π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Proposition 2.27[ 4, Coro. (3.8)]. IHJPAS. 36(2)2023 410 Let Ρ  be an Ζ¦-module and 𝑉 βŠ‚ Ρ . Then 𝑉 is EXNPQ2AB submodule of Ρ  if and only if for every ideals Ζ€1, Ζ€2, Ζ€3 of Ζ¦ and π‘₯ ∈ Ρ  such that Ζ€1Ζ€2Ζ€3π‘₯ βŠ† 𝑉 implies that either Ζ€1Ζ€2π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Proposition 2.28[ 4, Coro. (3.9)]. Let Ρ  be an Ζ¦-module and 𝑉 βŠ‚ Ρ . Then 𝑉 is EXNPQ2AB submodule of Ρ  if and only if for any π‘Ÿ, 𝑠 ∈ Ζ¦ and any ideal Ζ€ of Ζ¦ and every submodule 𝐴 of Ρ  with π‘Ÿπ‘ Ζ€π΄ βŠ† 𝑉 implies that either π‘Ÿπ‘ π΄ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘ŸΖ€π΄ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or 𝑠ƀ𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Proposition 2.29[ 4, Coro. (3.10)]. Let Ρ  be an Ζ¦-module and 𝑉 βŠ‚ Ρ . Then 𝑉 is EXNPQ2AB submodule of Ρ  if and only if for each π‘Ÿ ∈ Ζ¦ and any ideals Ζ€, 𝐽 of Ζ¦ and every submodule 𝐴 of Ρ  with π‘ŸΖ€π½π΄ βŠ† 𝑉 implies that either π‘ŸΖ€π΄ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Ÿπ½π΄ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or ƀ𝐽𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). 3. Main Results In this part we introduced some characterizations of Extend Nearly Pseudo Quasi-2-Absorbing submodules in multiplication modules. Proposition 3.1 Let Ρ  be a multiplication Ζ¦-module and 𝑉 β‰  Ρ . Then 𝑉 is EXNPQ2AB submodule of Ρ  if and only if whenever Σ‡1Σ‡2Σ‡3𝐴 βŠ† 𝑉 for some submodules Σ‡1,Σ‡2,Σ‡3, 𝐴 of Ρ , implies that either Σ‡1Σ‡2𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Σ‡1Σ‡3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Σ‡2Σ‡3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Proof. (⟹) Let Σ‡1Σ‡2Σ‡3𝐴 βŠ† 𝑉 for some submodules Σ‡1, Σ‡2, Σ‡3, 𝐴 of Ρ . Since Ρ  is a multiplication, then Σ‡1 = Ζ€1Ρ , Σ‡2 = Ζ€2Ρ , Σ‡3 = Ζ€3Ρ  and 𝐴 = Ζ€4Ρ  for some ideals Ζ€1, Ζ€2, Ζ€3 and Ζ€4of Ζ¦. That is Σ‡1Σ‡2Σ‡3𝐴 = Ζ€1Ζ€2Ζ€3(Ζ€4Ρ  ) βŠ† 𝑉. But 𝑉 is EXNPQ2AB submodule of Ρ , hence from Proposition 2.25 we get either Ζ€1Ζ€2(Ζ€4Ρ ) βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€3(Ζ€4Ρ ) βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3(Ζ€4Ρ ) βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Next, following either Σ‡1Σ‡2𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Σ‡1Σ‡3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Σ‡2Σ‡3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Ζ€ (⟸) Let Ζ€1Ζ€2Ζ€3𝐴 βŠ† 𝑉 for Ζ€1, Ζ€2, Ζ€3 are ideals of Ζ¦ and 𝐴 is a submodule of Ρ . Put Σ‡1 = Ζ€1Ρ , Σ‡2 = Ζ€2Ρ  and Σ‡3 = Ζ€3Ρ . That is Σ‡1Σ‡2Σ‡3𝐴 βŠ† 𝑉. Now, by hypotheses either Σ‡1Σ‡2𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Σ‡1Σ‡3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Σ‡2Σ‡3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ), thus Ζ€1Ζ€2𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3𝐴 βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Therefore by Proposition 2.25 𝑉 is EXNPQ2AB submodule of Ρ . Proposition 3.2 Let Ρ  be a multiplication Ζ¦-module and 𝑉 β‰  Ρ . Then 𝑉 is EXNPQ2AB submodule of Ρ  if and only if whenever Ζ‘1Ζ‘2Ζ‘3π‘₯ βŠ† 𝑉 for some submodules Ζ‘1, Ζ‘2, Ζ‘3 of Ρ ,π‘₯ ∈ Ρ , then either Ζ‘1Ζ‘2π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ‘1Ζ‘3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ‘2Ζ‘3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Proof. (⟹) Let 𝑉 is EXNPQ2AB submodule of Ρ  and Ζ‘1Ζ‘2Ζ‘3π‘₯ βŠ† 𝑉 for some submodules Ζ‘1,Ζ‘2, Ζ‘3 of Ρ  and π‘₯ ∈ Ρ . Since Ρ  is a multiplication, then Ζ‘1 = Ζ€1Ρ , Ζ‘2 = Ζ€2Ρ  and Ζ‘3 = Ζ€3Ρ  for some IHJPAS. 36(2)2023 411 ideals Ζ€1, Ζ€2and Ζ€3 of Ζ¦. That is Ζ‘1Ζ‘2Ζ‘3π‘₯ = Ζ€1Ζ€2Ζ€3π‘₯ βŠ† 𝑉. But 𝑉 is EXNPQ2AB submodule of Ρ , hence from Proposition 2.27 we get either Ζ€1Ζ€3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€2π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Next, following either Ζ‘1Ζ‘3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ‘2Ζ‘3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ‘1Ζ‘2π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). (⟸) Let Ζ€1Ζ€2Ζ€3π‘₯ βŠ† 𝑉 for Ζ€1, Ζ€2, Ζ€3 are ideals of Ζ¦ and π‘₯ ∈ Ρ . Put Ζ‘1 = Ζ€1Ρ , Ζ‘2 = Ζ€2Ρ  and Ζ‘3 = Ζ€3Ρ . That is Ζ‘1Ζ‘2Ζ‘3π‘₯ βŠ† 𝑉. Now, by hypotheses either Ζ‘1Ζ‘2π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ‘1Ζ‘3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ‘2Ζ‘3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ), thus Ζ€1Ζ€2π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3π‘₯ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Therefore by Proposition 2.27 𝑉 is EXNPQ2AB submodule of Ρ . Proposition 3.3 Let Ρ  be a multiplication Ζ¦-module and 𝑉 β‰  Ρ . Then 𝑉 is EXNPQ2AB submodule of Ρ  if and only if whenever Ι±1Ι±2Ι±3Σ‡ βŠ† 𝑉 for some Ι±1,Ι±2,Ι±3 ∈ Ρ , Σ‡ is a submodule of Ρ , implies that either Ι±1Ι±2Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ι±1Ι±3Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ι±2Ι±3Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Proof. (⟹) Let Ι±1Ι±2Ι±3Σ‡ βŠ† 𝑉 for some Ι±1,Ι±2,Ι±3 ∈ Ρ  and Σ‡ is a submodule of Ρ . That is (Ι±1)(Ι±2)(Ι±3)Σ‡ βŠ† 𝑉 Since Ρ  is a multiplication, then (Ι±1) = Ζ€1Ρ , (Ι±2) = Ζ€2Ρ , (Ι±3) = Ζ€3Ρ  and Σ‡ = Ζ€4Ρ  for some ideals Ζ€1, Ζ€2, Ζ€3 and Ζ€4 of Ζ¦. That is (Ι±1)(Ι±2)(Ι±3)Σ‡ = Ζ€1Ζ€2Ζ€3(Ζ€4Ρ ) βŠ† 𝑉. But 𝑉 is EXNPQ2AB submodule of Ρ , hence from Proposition 2.25 we get either Ζ€1Ζ€3(Ζ€4Ρ ) βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3(Ζ€4Ρ ) βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€2(Ζ€4Ρ ) βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Next, following either Ι±1Ι±3Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ι±2Ι±3Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ι±1Ι±2Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). (⟸) Let Ζ€1Ζ€2Ζ€3Σ‡ βŠ† 𝑉 for Ζ€1, Ζ€2, Ζ€3 are ideals of Ζ¦ and Σ‡ is a submodule of Ρ . Put (Ι±1) = Ζ€1Ρ , (Ι±2) = Ζ€2Ρ  and (Ι±3) = Ζ€3Ρ . That is (Ι±1)(Ι±2)(Ι±3)Σ‡ βŠ† 𝑉. That is Ι±1Ι±2Ι±3Σ‡ βŠ† 𝑉. Now, by hypotheses either Ι±1Ι±2Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ι±1Ι±3Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ι±2Ι±3Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ), thus (Ι±1)(Ι±2)Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or (Ι±1)(Ι±3)Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or (Ι±2)(Ι±3)Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Then Ζ€1Ζ€2Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€3Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3Σ‡ βŠ† 𝑉 + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Therefore by Proposition 2.25 𝑉 is EXNPQ2AB submodule of Ρ . Remark 3.4 If 𝑉 is an EXNPQ2AB submodule of an Ζ¦-module Ρ , then [𝑉:Ζ¦ Ρ ] need not to be EXNPQ2AB ideal of Ζ¦. The following example shows that: Let Ρ  = πš‰48 , Ζ¦ = πš‰ and the submodule 𝑉 = 〈16Μ…Μ…Μ…Μ… βŒͺ is EXNPQ2AB submodule of Ρ , since π‘ π‘œπ‘(πš‰48) = 〈2Μ…βŒͺ ∩ 〈3Μ…βŒͺ ∩ 〈8Μ…βŒͺ ∩ πš‰48 = 〈8Μ…βŒͺ and 𝐽(πš‰48) = 〈2Μ…βŒͺ ∩ 〈3Μ…βŒͺ = 〈6Μ…βŒͺ. Then 〈16Μ…Μ…Μ…Μ… βŒͺ + π‘ π‘œπ‘(πš‰48) + 𝐽(πš‰48) = 〈16Μ…Μ…Μ…Μ… βŒͺ + 〈8Μ…βŒͺ + 〈6Μ…βŒͺ = 〈2Μ…βŒͺ, hence for all Ι‘, Ι“, 𝑒 ∈ πš‰ and Ι± ∈ πš‰48 π‘ π‘’π‘β„Ž π‘‘β„Žπ‘Žπ‘‘ Ι‘Ι“π‘’π‘š ∈ 〈16Μ…Μ…Μ…Μ… βŒͺ, implies that either Ι‘Ι“Ι± ∈ 〈2Μ…βŒͺ or ɑ𝑒ɱ ∈ 〈2Μ…βŒͺ or ɓ𝑒ɱ ∈ 〈2Μ…βŒͺ. But [〈16Μ…Μ…Μ…Μ… βŒͺ:Ζ¦ πš‰48] = 16πš‰ is not an IHJPAS. 36(2)2023 412 EXNPQ2AB π‘–π‘‘π‘’π‘Žπ‘™ of πš‰, since 2.4.2.1 ∈ 16πš‰, for 1,2,4 ∈ πš‰, implies that 2.4.1 βˆ‰ 16πš‰ and 2.2.1 βˆ‰ 16πš‰ and 4.2.1 βˆ‰ 16πš‰. Under certain conditions, the above observation is fulfilled. Proposition 3.5 Let Ζ‘ β‰  Ρ  and Ρ  is faithful multiplication Ζ¦-module. Then Ζ‘ is EXNPQ2AB submodule of Ρ  if and only if [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. Proof. (⟹) Let Ζ‘ is EXNPQ2AB submodule of Ρ , and Ζ€1Ζ€2Ζ€3Ζ€4 βŠ† [Ζ‘:Ζ¦ Ρ ] for some ideals Ζ€1, Ζ€2, Ζ€3 and Ζ€4 of Ζ¦, then Ζ€1Ζ€2Ζ€3Ζ€4Ρ  βŠ† Ζ‘. But Ρ  is a multiplication, then Ζ€1Ζ€2Ζ€3Ζ€4Ρ  = Ζ‘1Ζ‘2Ζ‘3Ζ‘4 βŠ† Ζ‘, by taking Ζ€1Ρ  = Ζ‘1, Ζ€2Ρ  = Ζ‘2, Ζ€3Ρ  = Ζ‘3 and Ζ€4Ρ  = Ζ‘4. But Ζ‘ is EXNPQ2AB submodule of Ρ , then by Proposition 3.1 either Ζ‘1Ζ‘3Ζ‘4 βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ‘2Ζ‘3Ζ‘4 βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )or Ζ‘1Ζ‘2Ζ‘4 βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Since Ρ  is multiplication, then Ζ‘ = [Ζ‘:Ζ¦ Ρ ]Ρ , and since Ρ  is faithful multiplication, then by Lemma 2.6 π‘ π‘œπ‘(Ρ ) = π‘ π‘œπ‘(Ζ¦)Ρ  and by Lemma 2.7 𝐽(Ρ ) = 𝐽(Ζ¦)Ρ . Thus either Ζ€1Ζ€3Ζ€4Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or Ζ€2Ζ€3Ζ€4Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or Ζ€1Ζ€2Ζ€4Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ . Hence either Ζ€1Ζ€3Ζ€4 βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or Ζ€2Ζ€3Ζ€4 βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or Ζ€1Ζ€2Ζ€4 βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Therefore [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. (⟸) 𝑆uppose that [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦, and π‘Ÿπ‘ π‘‘π΄ βŠ† Ζ‘ for π‘Ÿ, 𝑠, 𝑑 ∈ Ζ¦ and 𝐴 is a submodule of Ρ , since Ρ  is a multiplication, then 𝐴 = Ζ€Ρ  for some ideal Ζ€ of Ζ¦, that is π‘Ÿπ‘ π‘‘Ζ€Ρ  βŠ† Ζ‘ , implies that π‘Ÿπ‘ π‘‘Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ], but [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦, then by Proposition 2.24 either π‘Ÿπ‘ Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or π‘Ÿπ‘‘Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or 𝑠𝑑ƀ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Thus either π‘Ÿπ‘ Ζ€Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or π‘Ÿπ‘‘Ζ€Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or 𝑠𝑑ƀѠ βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ . Since Ρ  is a faithful multiplication, then [Ζ‘:Ζ¦ Ρ ]Ρ  = Ζ‘ and by Lemma 2.6 and Lemma 2.7 either π‘Ÿπ‘ π΄ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Ÿπ‘‘π΄ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or 𝑠𝑑𝐴 βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Thus by Proposition 2.24 Ζ‘ is EXNPQ2AB submodule of Ρ . Proposition 3.6 Let Ζ‘ β‰  Ρ  and Ρ  is multiplication projective Ζ¦-module. Then Ζ‘ is EXNPQ2AB submodule of Ρ  if and only if [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. Proof. (⟹) Assume that Ζ‘ is EXNPQ2AB submodule of Ρ , and Ζ€1Ζ€2Ζ€3𝑏 βŠ† [Ζ‘:Ζ¦ Ρ ] for some ideals Ζ€1, Ζ€2, Ζ€3 of Ζ¦ and 𝑏 ∈ Ζ¦, then Ζ€1Ζ€2Ζ€3(𝑏Ѡ) βŠ† Ζ‘. But Ζ‘ is EXNPQ2AB submodule of Ρ , then by Proposition 2.25 either Ζ€1Ζ€3𝑏Ѡ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€2Ζ€3𝑏Ѡ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or Ζ€1Ζ€2𝑏Ѡ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Since Ρ  is multiplication, thenΖ‘ = [Ζ‘:Ζ¦ Ρ ]Ρ , and since Ρ  is projective Ζ¦-module Ρ , then by Lemma 2.10 π‘ π‘œπ‘(Ρ ) = π‘ π‘œπ‘(Ζ¦)Ρ  and by Lemma 2.9 𝐽(Ρ ) = 𝐽(Ζ¦)Ρ . Thus either Ζ€1Ζ€3𝑏Ѡ βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or Ζ€2Ζ€3𝑏Ѡ βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or Ζ€1Ζ€2𝑏Ѡ βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ . Hence Ζ€1Ζ€3𝑏 βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)or Ζ€2Ζ€3𝑏 βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or Ζ€1Ζ€2𝑏 βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Therefore by Proposition 2.27 [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. IHJPAS. 36(2)2023 413 (⟸) 𝑆uppose that [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦, and π‘Ÿπ‘ Ζ€π΄ βŠ† Ζ‘ for π‘Ÿ, 𝑠 ∈ Ζ¦ and some submodule 𝐴 of Ρ  and for some ideal Ζ€ of Ζ¦ since Ρ  is a multiplication, then 𝐴 = 𝐽Ѡ for some ideal 𝐽 of Ζ¦, that is π‘Ÿπ‘ Ζ€π½Ρ  βŠ† Ζ‘ , implies that π‘Ÿπ‘ Ζ€π½ βŠ† [Ζ‘:Ζ¦ Ρ ], but [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB π‘–π‘‘π‘’π‘Žπ‘™ of Ζ¦, then by Proposition 2.28 either π‘Ÿπ‘ π½ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or π‘ŸΖ€π½ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or 𝑠ƀ𝐽 βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Thus either π‘Ÿπ‘ π½Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or π‘ŸΖ€π½Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or 𝑠ƀ𝐽Ѡ βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ . Hence by Lemma 2.10 and Lemma 2.9 either π‘Ÿπ‘ π΄ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘ŸΖ€π΄ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or 𝑠ƀ𝐴 βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Thus by Proposition 2.28 Ζ‘ is EXNPQ2AB submodule of Ρ . Proposition 3.7 Let Ζ‘ β‰  Ρ  and Ρ  is non-singular multiplication Ζ¦-module Ρ  over an a good π‘Ÿπ‘–π‘›π‘” Ζ¦. Then Ζ‘ is EXNPQ2AB submodule of Ρ  if and only if [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. Proof. (⟹) Let π‘Žπ‘π‘π‘‘ ∈ [Ζ‘:Ζ¦ Ρ ] for π‘Ž, 𝑏, 𝑐, 𝑑 ∈ Ζ¦, then π‘Žπ‘π‘(𝑑Ѡ) βŠ† Ζ‘. But Ζ‘ is EXNPQ2AB submodule of Ρ , then by Proposition 2.24 either π‘Žπ‘(𝑑Ѡ) βŠ† Ζ‘ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘Žπ‘(𝑑Ѡ) βŠ† Ζ‘ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝑏𝑐(𝑑Ѡ) βŠ† Ζ‘ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). Since Ρ  is multiplication, then Ζ‘ = [Ζ‘:Ζ¦ Ρ ]Ρ  and since Ρ  is non-singular multiplication, then by Lemma 2.17 π‘ π‘œπ‘(Ρ ) = π‘ π‘œπ‘(Ζ¦)Ρ  and Ζ¦ is a good ring then by Remark 2.11 𝐽(Ρ ) = 𝐽(Ζ¦)Ρ . Thus either π‘Žπ‘(𝑑Ѡ) βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or (𝑑Ѡ) βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or π‘Žπ‘(𝑑Ѡ) βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ), then either π‘Žπ‘π‘‘ ∈ [Ζ‘:Ζ¦ Ρ ] + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or 𝑏𝑐𝑑 ∈ [Ζ‘:Ζ¦ Ρ ] + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or π‘Žπ‘π‘‘ ∈ [Ζ‘:Ζ¦ Ρ ] + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)). Hence by Proposition 2.24 [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. (⟸) 𝑆uppose that [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦, and π‘Žπ‘π‘π‘₯ ∈ Ζ‘ for π‘Ž, 𝑏, 𝑐 ∈ Ζ¦, π‘₯ ∈ Ρ , hence π‘Žπ‘π‘(π‘₯) βŠ† Ζ‘. Since Ρ  is a multiplication, then (π‘₯) = 𝐽Ѡ for some ideal 𝐽 of Ζ¦, that is π‘Žπ‘π‘π½Ρ  βŠ† Ζ‘ , implies that π‘Žπ‘π‘π½ βŠ† [Ζ‘:Ζ¦ Ρ ], but [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦, then by definition either π‘Žπ‘π½ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or π‘Žπ‘π½ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or 𝑏𝑐𝐽 βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Thus either π‘Žπ‘π½Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦) Ρ  + 𝐽(Ζ¦) Ρ  or π‘Žπ‘π½Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ] Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or 𝑏𝑐𝐽Ѡ βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ . Hence by Lemma 2.17 and Remark 2.11 either π‘Žπ‘(π‘₯) βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Žπ‘(π‘₯) βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or 𝑏𝑐(π‘₯) βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Next, follows either π‘Žπ‘π‘₯ ∈ Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Žπ‘π‘₯ ∈ Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or 𝑏𝑐π‘₯ ∈ Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ).Therefore Ζ‘ is EXNPQ2AB submodule of Ρ . As a direct application of Proposition 3.7, we get the following corollary: Corollary 3.8 Let Ζ‘ β‰  Ρ  and Ρ  is non-singular multiplication Ζ¦-module Ρ  over Artinian π‘Ÿπ‘–π‘›π‘” Ζ¦. Then Ζ‘ is EXNPQ2AB submodule of Ρ  if and only if [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. By Proof of Proposition 3.7 and using Lemma 2.15 we get: Proposition 3.9 Let Ζ‘ β‰  Ρ  and Ρ  is non-singular multiplication Ζ¦-module Ρ  over local ring Ζ¦. Then Ζ‘ is EXNPQ2AB submodule of Ρ  if and only if [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. IHJPAS. 36(2)2023 414 Proposition 3.10 Let Ζ‘ β‰  Ρ  and Ρ  is 𝑍-regular multiplication Ζ¦-module Ρ  over an a good ring Ζ¦. Then Ζ‘ is EXNPQ2AB submodule of Ρ  if and only if [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. Ζ€ Proof. (⟹) Let π‘Ÿπ‘ π‘‘Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ] for π‘Ÿ, 𝑠, 𝑑 ∈ Ζ¦ and Ζ€ is an ideal of Ζ¦, then π‘Ÿπ‘ π‘‘Ζ€Ρ  βŠ† Ζ‘. But Ζ‘ is EXNPQ2AB submodule of Ρ , then either π‘Ÿπ‘ Ζ€Ρ  βŠ† Ζ‘ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘Ÿπ‘‘Ζ€Ρ  βŠ† Ζ‘ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝑠𝑑ƀѠ βŠ† Ζ‘ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). Since Ρ  is multiplication, then Ζ‘ = [Ζ‘:Ζ¦ Ρ ]Ρ  and since Ρ  is a 𝑍-regular, then by Lemma 2.21 π‘ π‘œπ‘(Ρ ) = π‘ π‘œπ‘(Ζ¦)Ρ  and Ζ¦ is a good ring then by Remark 1.11 𝐽(Ρ ) = 𝐽(Ζ¦)Ρ . Thus either π‘Ÿπ‘ Ζ€Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or π‘Ÿπ‘‘Ζ€Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or 𝑠𝑑ƀѠ βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ), it follows that either π‘Ÿπ‘ Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ] + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or π‘Ÿπ‘‘Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ] + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or 𝑠𝑑ƀ βŠ† [Ζ‘:Ζ¦ Ρ ] + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)). Hence by Proposition 2.24 [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. (⟸) 𝑆uppose that [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦, and π‘Ÿπ‘ π‘‘π΄ βŠ† Ζ‘ for π‘Ÿ, 𝑠, 𝑑 ∈ Ρ  and 𝐴 is a submodule of Ρ . Since Ρ  is a multiplication, then 𝐴 = Ζ€Ρ , that is π‘Ÿπ‘ π‘‘π΄ = π‘Ÿπ‘ π‘‘Ζ€Ρ  βŠ† Ζ‘ , implies that π‘Ÿπ‘ π‘‘Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ], but [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB π‘–π‘‘π‘’π‘Žπ‘™ of Ζ¦, then by Proposition 2.24 either π‘Ÿπ‘ Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or π‘Ÿπ‘‘Ζ€ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or 𝑠𝑑ƀ βŠ† [Ζ‘:Ζ¦ Ρ ] + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦).Thus either π‘Ÿπ‘ Ζ€Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦) Ρ  + 𝐽(Ζ¦) Ρ  or π‘Ÿπ‘‘Ζ€Ρ  βŠ† [Ζ‘:Ζ¦ Ρ ] Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or 𝑠𝑑ƀѠ βŠ† [Ζ‘:Ζ¦ Ρ ]Ρ  + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ . Hence by Lemma 2.21 and Remark 2.11 either π‘Ÿπ‘ π΄ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Ÿπ‘‘π΄ βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or 𝑠𝑑𝐴 βŠ† Ζ‘ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Therefore Ζ‘ is EXNPQ2AB submodule of Ρ . As a direct application of Proposition 3.10, we get the following corollary: Corollary 3.11 Let Ζ‘ β‰  Ρ  and Ρ  is 𝑍-regular multiplication Ζ¦-module Ρ  over Artinian ring Ζ¦. Then Ζ‘ is EXNPQ2AB submodule of Ρ  if and only if [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. By Proof of Proposition 3.10 and using Lemma 2.15 we get: Proposition 3.12 Let Ζ‘ β‰  Ρ  and Ρ  𝑍-regular multiplication Ζ¦-module Ρ  over local ring Ζ¦. Then Ζ‘ is EXNPQ2AB submodule of Ρ  if and only if [Ζ‘:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. 4. More Result of EXNPQ2AB Submodules in Multiplication Modules. In this part we studied more result of EXNPQ2AB submodules in multiplication modules. And we got the most important results. Proposition 4.1 Let Ρ  be a finitely generated multiplication projective Ζ¦-module, and Ɓ is an ideal of Ζ¦ with π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ. Then Ɓ is EXNPQ2AB ideal of Ζ¦ if and only if ƁѠ is EXNPQ2AB submodule of Ρ . Proof. IHJPAS. 36(2)2023 415 (⟹) Let 𝐻1𝐻2𝐻3𝐴 βŠ† ƁѠ for some submodules 𝐻1,𝐻2,𝐻3, 𝐴 of Ρ . Since Ρ  is a multiplication, then 𝐻1 = πš₯1Ρ , 𝐻2 = πš₯2Ρ , 𝐻3 = πš₯3Ρ  and 𝐴 = πš₯4Ρ  for some ideals πš₯1, πš₯2, πš₯3 and πš₯4of Ζ¦. That is 𝐻1𝐻2𝐻3𝐴 = πš₯1πš₯2πš₯3πš₯4Ρ  βŠ† ƁѠ. But Ρ  is a finitely generated multiplication Ζ¦-module then by Lemma 2.18 πš₯1πš₯2πš₯3πš₯4 βŠ† Ɓ + π‘Žπ‘›π‘›Ζ¦(Ρ ), but π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ, implies that Ɓ + π‘Žπ‘›π‘›Ζ¦(Ρ ) = Ɓ, thus πš₯1πš₯2πš₯3πš₯4 βŠ† Ɓ. Now, by assumption Ɓ is EXNPQ2AB ideal of Ζ¦ then by Proposition 3.2 either πš₯1πš₯3πš₯4 βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or πš₯2πš₯3πš₯4 βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or πš₯1πš₯2πš₯4 βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)), hence either πš₯1πš₯3πš₯4Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or πš₯2πš₯3πš₯4Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or πš₯1πš₯2πš₯4Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ . Since Ρ  is a projective then by Lemma 2.10 and Lemma 2.9 (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) = (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ), thus either 𝐻1𝐻3𝐴 βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝐻2𝐻3𝐴 βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝐻1𝐻2𝐴 βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). Hence by Proposition 3.2 ƁѠ is EXNPQ2AB submodule of Ρ . (⟸) Let Ζ€1Ζ€2Ζ€3Ζ€4 βŠ† Ɓ, for Ζ€1, Ζ€2, Ζ€3 and Ζ€4 are ideals in Ζ¦, implies that Ζ€1Ζ€2Ζ€3(Ζ€4Ρ ) βŠ† ƁѠ. But ƁѠ is EXNPQ2AB submodule of Ρ , then by Proposition 2.25 either Ζ€1Ζ€2(Ζ€4Ρ ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ))or Ζ€1Ζ€3(Ζ€4Ρ ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or Ζ€2Ζ€3(Ζ€4Ρ ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). But Ρ  is a projective then (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) = (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ). Thus either Ζ€1Ζ€2Ζ€4Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or Ζ€1Ζ€3Ζ€4Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or Ζ€2Ζ€3Ζ€4Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ , hence either Ζ€1Ζ€2Ζ€4 βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or Ζ€1Ζ€3Ζ€4 βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or Ζ€2Ζ€3Ζ€4 βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Then by Proposition 2.25 Ɓ is EXNPQ2AB ideal of Ζ¦. Proposition 4.2 Let Ρ  be a faithful finitely generated multiplication Ζ¦-module, and Ɓ is an ideal of Ζ¦. Then Ɓ is EXNPQ2AB ideal of Ζ¦ if and only if ƁѠ is EXNPQ2AB submodule of Ρ . Proof. (⟹) Let π‘ŸΖ€π½π‘₯ βŠ† ƁѠ for any π‘Ÿ ∈ Ζ¦, π‘₯ ∈ Ρ  and Ζ€, 𝐽 are ideals of Ζ¦. Next, follows π‘ŸΖ€π½(π‘₯) βŠ† ƁѠ. Since Ρ  is a multiplication, then (π‘₯) = Ζ€1Ρ  for some ideal Ζ€1 of Ζ¦, that is π‘ŸΖ€π½Ζ€1Ρ  βŠ† ƁѠ. Thus by Lemma 2.18 we get π‘ŸΖ€π½Ζ€1 βŠ† Ɓ + π‘Žπ‘›π‘›(Ρ ), but Ρ  is faithful, then π‘Žπ‘›π‘›(Ρ ) = {0}, that is π‘ŸΖ€π½Ζ€1 βŠ† Ɓ. Since Ɓ is EXNPQ2AB ideal of Ζ¦, then by Proposition 2.27 either π‘ŸΖ€Ζ€1 βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or π‘Ÿπ½Ζ€1 βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or ƀ𝐽ƀ1 βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦), hence either π‘ŸΖ€Ζ€1Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or π‘Ÿπ½Ζ€1Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or ƀ𝐽ƀ1Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ , hence by Lemma 2.6 and Lemma 2.7 either π‘ŸΖ€(π‘₯) βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Ÿπ½(π‘₯) βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or ƀ𝐽(π‘₯) βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). That is either π‘ŸΖ€π‘₯ βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Ÿπ½π‘₯ βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or ƀ𝐽π‘₯ βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Hence by Proposition 2.26 ƁѠ is an EXNPQ2AB submodule of Ρ . (⟸) Let π‘Ÿπ‘ π‘‘Ζ€ βŠ† Ɓ for π‘Ÿ, 𝑠, 𝑑 ∈ Ζ¦ and Ζ€ ideal of Ζ¦, hence π‘Ÿπ‘ π‘‘(Ζ€Ρ ) βŠ† ƁѠ, but ƁѠ is an EXNPQ2AB submodule of Ρ , then either π‘Ÿπ‘ (Ζ€Ρ ) βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or π‘Ÿπ‘‘(Ζ€Ρ ) βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ) or 𝑠𝑑(Ζ€Ρ ) βŠ† ƁѠ + π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ ). Thus by Lemma 2.6 and Lemma 2.7 either π‘Ÿπ‘ Ζ€Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or π‘Ÿπ‘‘Ζ€Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or 𝑠𝑑ƀѠ βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ , hence either π‘Ÿπ‘ Ζ€ βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or π‘Ÿπ‘‘Ζ€ βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or 𝑠𝑑ƀ βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Therefore Ɓ is EXNPQ2AB ideal of Ζ¦. IHJPAS. 36(2)2023 416 Proposition 4.3 Let Ρ  be a finitely generated non-singular multiplication module over good ring Ζ¦ and Ɓ is an ideal of Ζ¦ with π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ. Then Ɓ is EXNPQ2AB ideal of Ζ¦ if and only if ƁѠ is EXNPQ2AB submodule of Ρ . Proof. (⟹) Let π‘Ÿπ‘ Ζ€π΄ βŠ† ƁѠ, for π‘Ÿ, 𝑠 ∈ Ζ¦, Ζ€ is an ideal of Ζ¦ and 𝐴 is a submodule of Ρ . Since Ρ  is a multiplication, then 𝐴 = Ζ€1Ρ , for some ideal Ζ€1 of Ζ¦, then π‘Ÿπ‘ Ζ€Ζ€1Ρ  βŠ† ƁѠ. But Ρ  is a finitely generated multiplication Ζ¦-module then by Lemma 2.18 π‘Ÿπ‘ Ζ€Ζ€1 βŠ† Ɓ + π‘Žπ‘›π‘›Ζ¦(Ρ ), since π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ, implies that Ɓ + π‘Žπ‘›π‘›Ζ¦(Ρ ) = Ɓ, hence π‘Ÿπ‘ Ζ€Ζ€1 βŠ† Ɓ. But Ɓ is EXNPQ2AB ideal of Ζ¦ then by Proposition 2.28 either π‘Ÿπ‘ Ζ€1 βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or π‘ŸΖ€Ζ€1 βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or 𝑠ƀƀ1 βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)). Thus either π‘Ÿπ‘ Ζ€1Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or π‘ŸΖ€Ζ€1Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or 𝑠ƀƀ1Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ). Since Ρ  is non-singular, then by Lemma 2.17 π‘ π‘œπ‘(Ζ¦)Ρ  = π‘ π‘œπ‘(Ρ ) and Ζ¦ is good ring then 𝐽(Ζ¦)Ρ  = 𝐽(Ρ ). Hence either π‘Ÿπ‘ Ζ€1Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘ŸΖ€Ζ€1Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝑠ƀƀ1Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). That is either π‘Ÿπ‘ π΄ βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘ŸΖ€π΄ βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝑠ƀ𝐴 βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). Therefore by Proposition 2.28 ƁѠ is EXNPQ2AB submodule of Ρ . (⟸) Let π‘ŸΖ€1Ζ€2Ζ€3 βŠ† Ɓ, for π‘Ÿ ∈ Ζ¦, and Ζ€1, Ζ€2, Ζ€3 are ideals of Ζ¦, implies that π‘ŸΖ€1Ζ€2(Ζ€3Ρ ) βŠ† ƁѠ. Since ƁѠ is EXNPQ2AB submodule of Ρ , then by Proposition 2.29 either π‘ŸΖ€1(Ζ€3Ρ ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘ŸΖ€2(Ζ€3Ρ ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or Ζ€1Ζ€2(Ζ€3Ρ ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). But Ρ  is non-singular and Ζ¦ is good ring then (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) = (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ). Thus either π‘ŸΖ€1Ζ€3Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or π‘ŸΖ€2Ζ€3Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or Ζ€1Ζ€2Ζ€3Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ), then either π‘ŸΖ€1Ζ€3 βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or π‘ŸΖ€2Ζ€3 βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or Ζ€1Ζ€2Ζ€3 βŠ† Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Hence by Proposition 2.28 Ɓ is EXNPQ2AB ideal of Ζ¦. Corollary 4.4 Let Ρ  be a finitely generated non-singular multiplication module over Artinian ring Ζ¦ and Ɓ is an ideal of Ζ¦ with π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ. Then Ɓ is EXNPQ2AB ideal of Ζ¦ if and only if ƁѠ is EXNPQ2AB submodule of Ρ . Proposition 4.5 Let Ρ  be a finitely generated non-singular multiplication module over local ring Ζ¦ and Ɓ is an ideal of Ζ¦ with π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ. Then Ɓ is EXNPQ2AB ideal of Ζ¦ if and only if ƁѠ is EXNPQ2AB submodule of Ρ . Proof. Similarly to the Proof of Proposition 4.3 by using Lemma 2.15. Proposition 4.6 Let Ρ  be a finitely generated multiplication 𝑍-regular module over good ring Ζ¦ and Ɓ is an ideal of Ζ¦ with π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ. Then Ɓ is EXNPQ2AB ideal of Ζ¦ if and only if ƁѠ is EXNPQ2AB submodule of Ρ . IHJPAS. 36(2)2023 417 Proof. (⟹) Let π‘Ÿπ‘ π‘‘π‘₯ ∈ ƁѠ for π‘Ÿ, 𝑠, 𝑑 ∈ Ζ¦ and π‘₯ ∈ Ρ , that is π‘Ÿπ‘ π‘‘βŒ©π‘₯βŒͺ βŠ† ƁѠ. Since Ρ  is a multiplication, then 〈π‘₯βŒͺ = Ζ€Ρ  for some ideal Ζ€ of Ζ¦, that is π‘Ÿπ‘ π‘‘Ζ€Ρ  βŠ† ƁѠ. But Ρ  is a finitely generated multiplication Ζ¦-module then by Lemma 2.18 π‘Ÿπ‘ π‘‘Ζ€ βŠ† Ɓ. But Ɓ is EXNPQ2AB ideal of Ζ¦ then by Proposition 2.24 either π‘Ÿπ‘ Ζ€ βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or π‘Ÿπ‘‘Ζ€ βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)) or 𝑠𝑑ƀ βŠ† Ɓ + (π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦)). Thus either π‘Ÿπ‘ Ζ€Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or π‘Ÿπ‘‘Ζ€Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ) or 𝑠𝑑ƀѠ βŠ† ƁѠ + (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ). Since Ρ  is 𝑍-π‘Ÿπ‘’π‘”π‘’π‘™π‘Žπ‘Ÿ then by Lemma 2.21 π‘ π‘œπ‘(Ζ¦)Ρ  = π‘ π‘œπ‘(Ρ ) and Ζ¦ is good ring then 𝐽(Ζ¦)Ρ  = 𝐽(Ρ ). Hence either π‘Ÿπ‘ Ζ€Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘Ÿπ‘‘Ζ€Ρ  βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝑠𝑑ƀѠ βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). That is either π‘Ÿπ‘ βŒ©π‘₯βŒͺ βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘Ÿπ‘‘βŒ©π‘₯βŒͺ βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘ π‘‘βŒ©π‘₯βŒͺ βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )), thus either π‘Ÿπ‘ π‘₯ ∈ ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘Ÿπ‘‘π‘₯ ∈ ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝑠𝑑π‘₯ ∈ ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). Therefore ƁѠ is EXNPQ2AB submodule of Ρ . (⟸) Let π‘Žπ‘π‘π‘‘ ∈ Ɓ, for π‘Ž, 𝑏, 𝑐, 𝑑 ∈ Ζ¦, implies that π‘Žπ‘π‘(𝑑Ѡ) βŠ† ƁѠ. Since ƁѠ is EXNPQ2AB submodule of Ρ , then by Proposition 2.24 either π‘Žπ‘(𝑑Ѡ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or π‘Žπ‘(𝑑Ѡ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) or 𝑏𝑐(𝑑Ѡ) βŠ† ƁѠ + (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )). But Ρ  is 𝑍-regular and Ζ¦ is good ring, then (π‘ π‘œπ‘(Ρ ) + 𝐽(Ρ )) = (π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ ). Thus either π‘Žπ‘π‘‘Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or π‘Žπ‘π‘‘Ρ  βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ  or 𝑏𝑐𝑑Ѡ βŠ† ƁѠ + π‘ π‘œπ‘(Ζ¦)Ρ  + 𝐽(Ζ¦)Ρ , then either π‘Žπ‘π‘‘ ∈ Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or π‘Žπ‘π‘‘ ∈ Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦) or 𝑏𝑐𝑑 ∈ Ɓ + π‘ π‘œπ‘(Ζ¦) + 𝐽(Ζ¦). Hence Ɓ is EXNPQ2AB ideal of Ζ¦. Corollary 4.7 Let Ρ  be a finitely generated multiplication 𝑍-regular module over Artinian ring Ζ¦ and Ɓ is an ideal of Ζ¦ with π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ. Then Ɓ is EXNPQ2AB ideal of Ζ¦ if and only if ƁѠ is EXNPQ2AB submodule of Ρ . Proposition 4.8 Let Ρ  be a finitely generated multiplication 𝑍-regular module over local ring Ζ¦ and Ɓ is an ideal of Ζ¦ with π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ. Then Ɓ is EXNPQ2AB ideal of Ζ¦ if and only if ƁѠ is EXNPQ2AB submodule of Ρ . Proof. Similar to the Proof of Proposition 4.6 by using Lemma 2.15. Proposition 4.9 Let Ρ  be a faithful finitely generated multiplication Ζ¦-module and 𝑉 β‰  Ρ , in which case the following claims are equivalent: 1. 𝑉 is EXNPQ2AB submodule of Ρ . 2. [𝑉:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. 3. 𝑉 = ƁѠ for some EXNPQ2AB ideal Ɓ of Ζ¦. Proof. (𝟏 ⇔ 𝟐) By Proposition 3.6. IHJPAS. 36(2)2023 418 (𝟐 β‡’ πŸ‘) Since [𝑉:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦ and Ρ  is a faithful, that is (0) = ɑ𝑛𝑛Ʀ(Ρ ) = [0:Ζ¦ Ρ ] βŠ† [𝑉:Ζ¦ Ρ ] and Ρ  is a multiplication, so V = [𝑉:Ζ¦ Ρ ]Ρ , implies that V = JΡ  for some EXNPQ2AB ideal J = [𝑉:Ζ¦ Ρ ] of Ζ¦. (πŸ‘ β‡’ 𝟐) Suppose that 𝑉 = 𝐽Ѡ for some EXNPQ2AB ideal 𝐽 of Ζ¦. Since Ρ  is multiplication, then V = [V:Ζ¦ Ρ ]Ρ . That is 𝐽Ѡ = [V:Ζ¦ Ρ ]Ρ , but Ρ  is faithful finitely generated multiplication then by Lemma 2.23 we get [V:Ζ¦ Ρ ] = J. Thus [V:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. Proposition 4.10 Let Ρ  be a finitely generated multiplication projective Ζ¦-module and 𝑉 β‰  Ρ , in which case the following claims are equivalent: 1. 𝑉 is EXNPQ2AB submodule of Ρ . 2. [𝑉:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. 3. 𝑉 = ƁѠ for some EXNPQ2AB ideal Ɓ of Ζ¦. Proof. (𝟏 ⇔ 𝟐) By Proposition 3.7. (𝟐 β‡’ πŸ‘) Clear. (πŸ‘ β‡’ 𝟐) Assume that 𝑉 = ƁѠ …..(1) for some EXNPQ2AB ideal Ɓ of Ζ¦ with π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ. while Ρ  is a multiplication, then 𝑉 = [𝑉:Ζ¦ Ρ ]Ѡ…..(2), from (1) and (2) we have [𝑉:Ζ¦ Ρ ]Ρ  = ƁѠ. Since Ρ  is a finitely generated, then by Lemma 2.22 Ρ  is weak cancellation, then [𝑉:Ζ¦ Ρ ] + π‘Žπ‘›π‘›Ζ¦(Ρ ) = Ɓ + π‘Žπ‘›π‘›Ζ¦(Ρ ), but π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† Ɓ, and π‘Žπ‘›π‘›Ζ¦(Ρ ) βŠ† [𝑉:Ζ¦ Ρ ], implies that π‘Žπ‘›π‘›Ζ¦(Ρ ) + Ɓ = Ɓ and [𝑉:Ζ¦ Ρ ] + π‘Žπ‘›π‘›Ζ¦(Ρ ) = [𝑉:Ζ¦ Ρ ]. Thus Ɓ = [𝑉:Ζ¦ Ρ ], but Ɓ is EXNPQ2AB ideal of Ζ¦, hence [𝑉:Ζ¦ Ρ ] is EXNPQ2AB ideal of Ζ¦. 5. Conclusion. In this paper, we introduced the some characterizations in class of multiplication modules. 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