IHJPAS. 36 (3) 2023 306 This work is licensed under a Creative Commons Attribution 4.0 International License *Corresponding Author: mohmad.e.dahash35391@st.tu.edu.iq Abstract Let โ„‹ be a moduleover a commutativering ๐‘… with identity.Before studying the notion of Strongly Pseudo Nearly Semi-2-Absorbing submodule, where a propersubmodule โ„ฑ of an ๐‘…- module โ„‹ is called to be Strongly Pseudo Nearly Semi-2-Absorbingsubmodule of โ„‹ if โˆ€ ๐‘ข2ฯฐ โˆˆ โ„ฑ, for ๐‘ข โˆˆ ๐‘…, ๐œ˜ โˆˆ โ„‹ it follows that either ๐‘ข๐œ˜ โˆˆ โ„ฑ + ๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) or ๐‘ข2 โˆˆ [โ„ฑ + ๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹):๐‘… โ„‹], we need to mention the ideal [โ„ฑ โ„‹๐‘… : ] = {๐‘Ÿ โˆˆ ๐‘…: ๐‘Ÿโ„‹ โŠ† โ„ฑ} and the basics that you need to study the notion of Strongly Pseudo Nearly Semi-2-Absorbing submodule. Also we introduce several characterzations of Strongly Pseudo Nearly Semi-2-Absorbing submodule in classes of multiplication modules and other types of modules. We also had no luck the ideal [โ„ฑ โ„‹๐‘… : ] = {๐‘Ÿ โˆˆ ๐‘…: ๐‘Ÿโ„‹ โŠ† โ„ฑ} is not Strongly Pseudo Nearly Semi-2-Absorbing ideal. Also noted that [โ„ฑ โ„‹๐‘… : ] is Strongly Pseudo Nearly Semi-2-Absorbing ideal under several conditions. Also we introduce the characterization of the notion of Strongly Pseudo Nearly Semi-2-Absorbing ideals by special kind of submodules. Keywords: STPNS-Asubmodules, STPNS-Aideal, faithful module, projective module, Z-regular module. 1. Introduction In this part we note that the ideal [โ„ฑ โ„‹๐‘… : ] = {๐‘Ÿ โˆˆ ๐‘…: ๐‘Ÿโ„‹ โŠ† โ„ฑ} is not STPNS-Aideals and we gave an example of that. We also noted that [โ„ฑ โ„‹๐‘… : ] is STPNS-Aideals under several conditions, which was previously by several researchers an is the first condition faithful module submitted by researcher kach in 1982. The second condition projective module was also presented by the same researcher. The third condition is two conditions combined together Z-regular module and content module, which was presented in (1973, 1989). The quarter condition is two conditions combined together non-singular module and content module was presented in (1976, 1989). Also in this part doi.org/10.30526/36.3.3065 Article history: Received 9 October 2022, Accepted 28 November 2022, Published in July 2023. Ibn Al-Haitham Journal for Pure and Applied Sciences Journal homepage: jih.uobaghdad.edu.iq Strongly Pseudo Nearly Semei-2-Absorbing submodule(๐ˆ๐ˆ) *Mohmad E. Dahash Department of Mathematics College of Computer Science and Mathematics Tikrit University, Iraq. mohmad.e.dahash35391@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:mohmad.e.dahash35391@st.tu.edu.iq mailto:mohmad.e.dahash35391@st.tu.edu.iq mailto:mohmad.e.dahash35391@st.tu.edu.iq mailto:H.mohammadali@tu.edu.iq mailto:H.mohammadali@tu.edu.iq mailto:H.mohammadali@tu.edu.iq IHJPAS. 36 (3) 2023 307 we introduce the characterization of the concept of STPNS-Aideals by special kind of submodules. 2. Basic Properties In this part we introduce the basic properties of the concept Strongly Pseudo Nearly Semei-2- Absorbing submodules. Definition 2.1 [1]. [โ„ฑ โ„‹๐‘… : ] = {๐‘Ÿ โˆˆ ๐‘…: ๐‘Ÿโ„‹ โŠ† โ„ฑ} where โ„ฑ is a submodule of an ๐‘…-module โ„‹. Lemma 2.2 [2, prop.(3.3)]. Let โ„‹ an โ„›-module and โ„ฑ โŠ‚ โ„‹. Then โ„ฑ is a STPNS-2-A submodule of โ„‹ ifandonlyif ฮ™2โ„’ โŠ† โ„ฑ for ฮ™ is an ideal of ๐‘… and โ„’ โŠ† โ„‹ it means that either ฮ™โ„’ โŠ† โ„ฑ + (J(โ„‹) โˆฉ soc(โ„‹)) or ฮ™2 โŠ† [โ„ฑ + (J(โ„‹) โˆฉ soc(โ„‹)):โ„› โ„‹]. Definition 2.3 [3]. AnRโ€“module โ„‹ is called to befaithful if ๐ด๐‘›๐‘›๐‘… (โ„‹) = (0) . ๐‘Šโ„Ž๐‘’๐‘Ÿ๐‘’ ๐ด๐‘›๐‘› ( โ„‹) = { ๐‘Ÿ ๏ƒŽ ๐‘… โˆถ ๐‘Ÿ โ„‹ = (0) }. Definition 2.4 [4]. AnRโ€“module โ„‹ is called to bemultiplication , if any submodule โ„ฑ of โ„‹ is of theform โ„ฑ = ๐ผ โ„‹ for some ideal ๐ผ of ๐‘… . Equivalent to โ„ฑ = [โ„ฑ โ„‹๐‘… : ]โ„‹. Lemma 2.5 [5 , coro. ( 2 .1. 14 ) (i)]. Let โ„‹ be a faithfulmultiplication Rโ€“module , then ๐‘†๐‘œ๐‘ ( โ„‹ ) = ๐‘†๐‘œ๐‘ ( ๐‘… ) โ„‹. Lemma 2.6 [6]. Let โ„‹ be a faithfulmultiplication Rโ€“module , then ๐ฝ ( โ„‹ ) = ๐ฝ ( ๐‘… ) โ„‹. Lemma 2.7 [2, coro.(3.4)]. Let โ„‹ an ๐‘…-module and โ„ฑ โŠ‚ โ„‹. Then โ„ฑ is STPNS-2-A submodule of โ„‹ ifandonlyif u2โ„’ โŠ† โ„ฑ for ๐‘ข โˆˆ ๐‘… and โ„’ โŠ† โ„‹ it means that either ๐‘ขโ„’ โŠ† โ„ฑ + J(โ„‹) โˆฉ soc(โ„‹) or u2 โˆˆ [โ„ฑ + J(โ„‹) โˆฉ soc(โ„‹):โ„› โ„‹]. Definition 2.8 [3]. An Rโ€“module โ„‹ isaprojective if for any Rโ€“epimorphism ๐‘“ โˆถ ๐œ‡ ๏‚ฎ ๐œ‡๏‚ข and every R โ€“ homomorphism ๐‘” โˆถ โ„‹ ๏‚ฎ ๐œ‡๏‚ข , there exists an Rโ€“homomorphism โ„Ž โˆถ โ„‹ ๏‚ฎ ๐œ‡ such that the followingdiagram is commutethat is ๐‘“๐‘œโ„Ž = ๐‘”. Lemma 2.9 [6 , prop. ( 3 . 24 )] Let โ„‹ beaprojective Rโ€“module, then ๐‘†๐‘œ๐‘ ( โ„‹ ) = ๐‘†๐‘œ๐‘ ( ๐‘… ) โ„‹. Lemma 2.10 [3 , Theo. (9 . 2.1) (a)] For anyprojective Rโ€“module โ„‹, we have ๐ฝ ( โ„‹) = ๐ฝ ( ๐‘… ) โ„‹. Definition 2.11 [7] An Rโ€“module โ„‹ is calledZโ€“regular if for any ๐‘ฅ ๏ƒŽ โ„‹ โˆƒ ๐‘” ๏ƒŽ โ„‹ = HomR(โ„‹ , ๐‘… ) such that ๐‘ฅ = ๐‘” (๐‘ฅ) ๐‘ฅ. Lemma 2.12 [6 , proposition (3 . 25)] Let โ„‹ be a Z โ€“ regularRโ€“module then ๐‘†๐‘œ๐‘ ( โ„‹ ) = ๐‘†๐‘œ๐‘ ( ๐‘… ) โ„‹. Definition 2.13 [8]. An R-module โ„‹ is said to be content module if (โˆฉ๐‘–โˆˆ๐ผ ๐ด๐‘– )โ„‹ =โˆฉ๐‘–โˆˆ๐ผ ๐ด๐‘– โ„‹ for each family of ideals ๐ด๐‘– in ๐‘…. IHJPAS. 36 (3) 2023 308 Lemma 2.14 [6 , proposition (1 . 11)]. If โ„‹ is content module, then ๐ฝ ( โ„‹) = ๐ฝ ( ๐‘… ) โ„‹. Definition 2.15 [9]. ๐‘(โ„‹) = {๐‘ฅ โˆˆ โ„‹: ๐‘Ž๐‘›๐‘›(๐‘ฅ)๐‘’๐‘ ๐‘ ๐‘’๐‘›๐‘ก๐‘–๐‘Ž๐‘™ ๐‘–๐‘‘๐‘’๐‘Ž๐‘™ ๐‘–๐‘› ๐‘…} is called singular submodule of โ„‹. If ๐‘(โ„‹) = โ„‹, then โ„‹ is called the singular module . If ๐‘(โ„‹) = 0, then โ„‹ is called the non-singular module. Lemma 2.16 [9 , coro. ( 1 . 26 )]. Let โ„‹ be a non-singularRโ€“module , then ๐‘†๐‘œ๐‘ ( โ„‹ ) = ๐‘†๐‘œ๐‘ ( ๐‘… ) โ„‹. Definition 2.17 [3]. An R โ€“ module โ„‹ is finitely generated if โ„‹ = โŒฉ๐‘ง1, ๐‘ง2, ๐‘ง3, . . . , ๐‘ง๐‘›โŒช = ๐‘…๐‘ง1, ๐‘…๐‘ง2, ๐‘…๐‘ง3, . . . ๐‘…๐‘ง๐‘›, where ๐‘ง1, ๐‘ง2, ๐‘ง3, . . . , ๐‘ง๐‘› โˆˆ โ„‹. Lemma 2.18 [10 ,coro. of theo. ( 9 )] Let โ„‹ be a finitelygenerated multiplicationRโ€“module and I , J ideals of R . Then I โ„‹ ๏ƒ Jโ„‹ ifandonlyif I ๏ƒ J + annR( โ„‹ ). Definition 2.19 [11]. AnRโ€“module โ„‹ is called to becancellation ifwhenever I โ„‹ = J โ„‹ for any ideals I, J of ๐‘…, implies that I = J. Lemma 2.20 [ 11 , prop. ( 3 . 1)]. If โ„‹ is a multiplicationRโ€“module, then โ„‹ iscancellation ifandonlyif โ„‹ is a faithfulfinitely generated. 3. The Results Inthis section we introduce the definition of Strongly Pseudo Nearly Semi-2-Absorbingsubmodule and we introduce several characterzations of STPNS-Asubmodules in classes of multiplicationmodules and other types of modules: Definition 3.1[2] A propersubmodule โ„ฑ of an ๐‘…-module โ„‹ is called to be Strongly Pseudo Nearly Semi-2- Absorbing submodule of โ„‹ (for short STPNS) if whenever ๐‘ข2ฯฐ โˆˆ โ„ฑ, for ๐‘ข โˆˆ ๐‘…, ๐œ˜ โˆˆ โ„‹ implies that either ๐‘ข๐œ˜ โˆˆ โ„ฑ + ๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) or ๐‘ข2 โˆˆ [โ„ฑ + ๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹):๐‘… โ„‹]. The following proposition gives characterization of STPNS-Asubmodules in classes of multiplication modules. Proposition 3.2 A propersubmodule ๐น of a multiplication๐‘…-module H is STPNS-Asubmodule of ๐ป if and only if ๐ด2๐พ โŠ† ๐น for ๐ด, ๐พ are submodules of ๐ป, implies that either ๐ด๐พ โŠ† ๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)) or ๐ด2 โŠ† ๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)). . Proof (โŸน) Let ๐‘Ÿ2๐พ โŠ† ๐น for ๐‘Ÿ โˆˆ ๐‘…, ๐พ โŠ† ๐ป. But ๐ป is a multiplicationmodule, then ๐พ= ๐ผ๐ป forsome ideal I of ๐‘…, it follows that ๐‘Ÿ2๐ผ๐ป โŠ† ๐น, it follows by hypothesis either ๐‘Ÿ๐ผ๐ป โŠ† ๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป))or ๐‘Ÿ2 โˆˆ [๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)) :๐‘… ๐ป]. That is either ๐‘Ÿ๐พ โŠ† ๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)) or ๐‘Ÿ 2 โˆˆ [๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)) :๐‘… ๐ป]. Hence ๐น is STPNS-Asubmodule of ๐ป. (โŸธ)Let ๐ด2๐พ โŠ† ๐น for ๐ด, ๐พ are submodules of a multiplication module ๐ป, it follows that (๐ผ๐ป)2(๐ฝ๐ป) = ๐ผ2๐ฝ๐ป โŠ† ๐น for some ideals ๐ผ, ๐ฝ of ๐‘…. Since ๐น is STPNS-Asubmodule of ๐ป, then by IHJPAS. 36 (3) 2023 309 lemma 2.2 we have either ๐ผ๐ฝ๐ป โŠ† ๐น + ๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป) or ๐ผ2 โŠ† [๐น + ๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป) :๐‘… ๐ป], that is either ๐ด๐พ โŠ† ๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)) or ๐ด2 โŠ† ๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)). As a directresult of proposition 3.2 we have the followingcorollaries. Corollary 3.3 A propersubmodule ๐น of a multiplication ๐‘…-module ๐ป is STPNS-Asubmodule of ๐ป if and only if ๐ด2โ„Ž โŠ† ๐น for ๐ด is a submodules of ๐ป and โ„Ž โˆˆ ๐ป, it means that either ๐ดโ„Ž โŠ† ๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)) or ๐ด2 โŠ† ๐น + (๐ฝ(๐ป) โˆฉ ๐‘ ๐‘œ๐‘(๐ป)). Corollary 3.4 A propersubmodule ๐น of a multiplication๐‘…-module ๐ป is STPNS-Asubmodule of ๐ป if and only if โ„Ž2๐พ โŠ† ๐น for โ„Ž โˆˆ ๐ป and ๐พ โŠ† ๐ป, it means that either โ„Ž๐พ โŠ† ๐น + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)) or โ„Ž2 โŠ† ๐น + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)). Remaek 3.5 If โ„ฑ is an STPNS-Asubmodule of an R_module โ„‹, then [โ„ฑ โ„‹๐‘… : ] it doesnโ€™t have to be an STPNS- Aideal of R, the followingexample explainthat: Considerthe Z-module ๐‘36, the submodule โ„ฑ = โŒฉ12ฬ…ฬ…ฬ…ฬ… โŒช is an STPNS-Asubmodule of the Z_module ๐‘36, since 2 2. 3ฬ… โˆˆ โŒฉ12ฬ…ฬ…ฬ…ฬ… โŒช, implies that 2. 3ฬ… โˆˆ โŒฉ12ฬ…ฬ…ฬ…ฬ… โŒช + (๐ฝ(๐‘36) โˆฉ ๐‘ ๐‘œ๐‘(๐‘36)) = โŒฉ12ฬ…ฬ…ฬ…ฬ… โŒช + (โŒฉ6ฬ…โŒช โˆฉ โŒฉ6ฬ…โŒช) = โŒฉ6ฬ…โŒช, but [โŒฉ12ฬ…ฬ…ฬ…ฬ… โŒช ๐‘36๐‘… : ] = 12๐‘ is not to be an STPNS-Aideal of Z, because 22. 3 โˆˆ 12๐‘,but 2.3 โˆ‰ 12๐‘ + (๐‘ ๐‘œ๐‘(๐‘) โˆฉ ๐ฝ(๐‘)) = 12๐‘ and22 โˆ‰ [12๐‘ + (๐ฝ(๐‘) โˆฉ ๐‘ ๐‘œ๐‘(๐‘)) ๐‘… : ๐‘] = 12๐‘. The above remark satsfay under certain conditions. Proposition 3.6 Let โ„‹ a faithful multiplication ๐‘…-module and โ„ฑ โŠ‚ โ„‹.Then โ„ฑ is STPNS-Asubmodule of โ„‹ if and only if [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of ๐‘…. Proof (โŸน) Let I2J โŠ† [โ„ฑ:R โ„‹] forsome ideals I, J of R, hence I 2(Jโ„‹) โŠ† โ„ฑ. But โ„ฑ is STPNS- Asubmodule of โ„‹, then by lemma 2.2 either I(Jโ„‹) โŠ† โ„ฑ + (J(โ„‹) โˆฉ soc(โ„‹))or I2 โŠ† [โ„ฑ + (J(โ„‹) โˆฉ soc(โ„‹)):R โ„‹]. Since โ„‹ is multiplication, then โ„ฑ = [โ„ฑ:R โ„‹]โ„‹ and since โ„‹ is faithful multiplication, then by lemmas 2.5, 2.6 soc(โ„‹) = soc(R)โ„‹ and J(โ„‹) = J(R)โ„‹. Thus either I(Jโ„‹) โŠ† [โ„ฑ:R โ„‹]โ„‹ + J(R)โ„‹ โˆฉ soc(R)โ„‹ or I 2โ„‹ โŠ† [โ„ฑ:R โ„‹]โ„‹ + (J(R)โ„‹ โˆฉ soc(R)โ„‹), thus either IJ โŠ† [โ„ฑ:R โ„‹] + (J(R) โˆฉ soc(R)) or I 2 โŠ† [โ„ฑ:R โ„‹] + (J(R) โˆฉ soc(R)) = [[โ„ฑ:R โ„‹] + (J(R) โˆฉ soc(R)):R R]. Hence [โ„ฑ:R โ„‹] is STPNS-Aideal of R. (โŸธ) Let A2L โŠ† โ„ฑ for A, L aresubmodules of โ„‹. Since โ„‹ isamultiplication, then A = Iโ„‹and L = Jโ„‹ forsome ideals I, J of R, that is (Iโ„‹)2Jโ„‹ โŠ† โ„ฑ, implies that I2J โŠ† [โ„ฑ:R โ„‹], but [โ„ฑ:R โ„‹] is STPNS-Aideal of R, theneither IJ โŠ† [โ„ฑ:R โ„‹] + (J(R) โˆฉ soc(R)) or I 2 โŠ† [[โ„ฑ:R โ„‹] + (J(R) โˆฉ soc(R)) :R R] = [โ„ฑ:R โ„‹] + (J(R) โˆฉ soc(R)),thus either IJโ„‹ โŠ† [โ„ฑ:R โ„‹]โ„‹ + (J(R)โ„‹ โˆฉ soc(R)โ„‹) or I2โ„‹ โŠ† [โ„ฑ:R โ„‹]โ„‹ + (J(R)โ„‹ โˆฉ soc(R)โ„‹). Hence by lemmaies 2.5, 2.6 either AL โŠ† โ„ฑ + (J(โ„‹) โˆฉ soc(โ„‹)) or A2 โŠ† [โ„ฑ + (J(โ„‹) โˆฉ soc(โ„‹)):R โ„‹]. Thus by proposition 3.2 โ„ฑ is STPNS-Asubmodule of โ„‹. Proposition 3.7 Let โ„‹ be a multiplication projectiveRโ€“module and โ„ฑ โŠ‚ โ„‹ . Then โ„ฑ is STPNS-Asubmodule of โ„‹ if and only if [โ„ฑ:๐‘… โ„‹]is STPNS-Aideal of R. Proof (โŸน) Let ๐‘ข2๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹] for some ๐‘ข โˆˆ ๐‘… and ๐ฝ is an ideals of ๐‘…, hence ๐‘ข 2(๐ฝโ„‹) โŠ† โ„ฑ. But โ„ฑ is STPNS -2-Absorbing submodule of โ„‹, then by lemma 2.7 either ๐‘ข(๐ฝโ„‹) โŠ† โ„ฑ + (๐ฝ(โ„‹) โˆฉ IHJPAS. 36 (3) 2023 310 ๐‘ ๐‘œ๐‘(โ„‹))or ๐‘ข2 โŠ† [โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)):๐‘… โ„‹]. Since โ„‹ is multiplication, then โ„ฑ = [โ„ฑ:๐‘… โ„‹]โ„‹ and since โ„‹ is projective, then by lemmaies 2.9, 2.10 ๐‘ ๐‘œ๐‘(โ„‹) = ๐‘ ๐‘œ๐‘(๐‘…)โ„‹ and ๐ฝ(โ„‹) = ๐ฝ(๐‘…)โ„‹. Thus either ๐‘ข(๐ฝโ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + ๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹ or ๐‘ข 2โ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹), thus either ๐‘ข๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐‘ข 2 โŠ† [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) = [[โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…]. Hence [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of ๐‘…. (โŸธ) Let ๐ด2โ„Ž โŠ† โ„ฑ for ๐ด is a submodules of โ„‹ and โ„Ž โˆˆ โ„‹. But โ„‹ is a multiplication, then ๐ด = ๐ผโ„‹ and โ„Ž = ๐ฝโ„‹ for some ideals ๐ผ, J of ๐‘…, that is (๐ผโ„‹)2๐ฝโ„‹ โŠ† โ„ฑ, implies that ๐ผ2๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹], but [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of ๐‘…, then either ๐ผ๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐ผ 2 โŠ† [[โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) :๐‘… ๐‘…] = [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) , thus either ๐ผ๐ฝโ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐ผ 2โ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). Hence by lemmaies 2.9, 2.10 either ๐ดโ„Ž โŠ† โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)) or ๐ด2 โŠ† [โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)):๐‘… โ„‹]. Thus by corollary 3.3 โ„ฑ is STPNS-Asubmodule of โ„‹. Proposition 3.8 Let โ„‹ be a multiplicationZ-Regular content Rโ€“module and โ„ฑ be a propersubmdule of โ„‹ . Then โ„ฑ is STPNS-Asubmodule of โ„‹ if and only if [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . Proof (โŸน) Let ๐‘Ž2๐‘ โˆˆ [โ„ฑ:๐‘… โ„‹] for some ๐‘Ž, ๐‘ โˆˆ ๐‘…, hence ๐‘Ž 2(๐‘โ„‹) โŠ† โ„ฑ. But โ„ฑ is STPNS -2-Absorbing submodule of โ„‹, then by lemma 2.7 either ๐‘Ž(๐‘โ„‹) โŠ† โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)) or ๐‘Ž2 โˆˆ [โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)):๐‘… โ„‹]. Since โ„‹ is multiplication, then โ„ฑ = [โ„ฑ:๐‘… โ„‹]โ„‹ and since โ„‹ is Z- Regular content Rโ€“module, then by lemmas 2.12, 2.14 ๐‘ ๐‘œ๐‘(โ„‹) = ๐‘ ๐‘œ๐‘(๐‘…)โ„‹ and ๐ฝ(โ„‹) = ๐ฝ(๐‘…)โ„‹. Thus either ๐‘Ž(๐‘โ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + ๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹ or ๐‘Ž 2โ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹), thus either ๐‘Ž๐‘ โˆˆ [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐‘Ž 2 โˆˆ [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) = [[โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…]. Hence [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of ๐‘…. (โŸธ) Let โ„Ž2๐ด โŠ† โ„ฑ for โ„Ž โˆˆ โ„‹ and ๐ด is a submodules of โ„‹. But โ„‹ is a multiplication, then โ„Ž = ๐ผโ„‹ and ๐ด = ๐ฝโ„‹ forsome ideals ๐ผ, J of ๐‘…, that is (๐ผโ„‹)2๐ฝโ„‹ โŠ† โ„ฑ, implies that ๐ผ2๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹], but [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of ๐‘…, then either ๐ผ๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐ผ 2 โŠ† [[โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) :๐‘… ๐‘…] = [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) , thus either ๐ผ๐ฝโ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐ผ 2โ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). Hence lemmas 2.12, 2.14 ๐‘ ๐‘œ๐‘(โ„‹) = ๐‘ ๐‘œ๐‘(๐‘…)โ„‹ and ๐ฝ(โ„‹) = ๐ฝ(๐‘…)โ„‹ either โ„Ž๐ด โŠ† โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)) or โ„Ž2 โŠ† [โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)):๐‘… โ„‹]. Thus by corollary 3.4 โ„ฑ is STPNS-Asubmodule of โ„‹. Proposition 3.9 Let โ„‹ be a multiplicationnon-singular contentRโ€“module and โ„ฑ be a propersubmdule of โ„‹ . Then โ„ฑ is STPNS-Asubmodule of โ„‹ ifandonlyif [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R. Proof (โŸน) Let ๐ผ2๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹] for some ideals ๐ผ, ๐ฝ of ๐‘…, hence ๐ผ 2(๐ฝโ„‹) โŠ† โ„ฑ. But โ„ฑ is STPNS -2- Absorbing submodule of โ„‹, then by lemma 2.2 either ๐ผ(๐ฝโ„‹) โŠ† โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹))or ๐ผ2 โŠ† [โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)):๐‘… โ„‹]. Since โ„‹ is multiplication, then โ„ฑ = [โ„ฑ:๐‘… โ„‹]โ„‹ and since โ„‹ non- singular content Rโ€“module, then by lemmas 2.16, 2.14 ๐‘ ๐‘œ๐‘(โ„‹) = ๐‘ ๐‘œ๐‘(๐‘…)โ„‹ and ๐ฝ(โ„‹) = ๐ฝ(๐‘…)โ„‹. Thus either ๐ผ(๐ฝโ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + ๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹ or ๐ผ 2โ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹), thus either ๐ผ๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐ผ 2 โŠ† [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) = [[โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…]. Hence [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of ๐‘…. (โŸธ) Let ๐ด2๐ฟ โŠ† โ„ฑ for ๐ด, ๐ฟ are submodules of โ„‹. Since โ„‹ is a multiplication, then ๐ด = ๐ผโ„‹and ๐ฟ = ๐ฝโ„‹ for some ideals ๐ผ, ๐ฝ of ๐‘…, that is (๐ผโ„‹)2๐ฝโ„‹ โŠ† โ„ฑ, implies that ๐ผ2๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹], but [โ„ฑ:๐‘… โ„‹] IHJPAS. 36 (3) 2023 311 is STPNS-Aideal of ๐‘…, then either ๐ผ๐ฝ โŠ† [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐ผ 2 โŠ† [[โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) :๐‘… ๐‘…] = [โ„ฑ:๐‘… โ„‹] + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) , thus either ๐ผ๐ฝโ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐ผ2โ„‹ โŠ† [โ„ฑ:๐‘… โ„‹]โ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). Hence by lemmas2.16, 2.14 either ๐ด๐ฟ โŠ† โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)) or ๐ด2 โŠ† [โ„ฑ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹)):๐‘… โ„‹]. Thus by proposition 3.2 โ„ฑ is STPNS-Asubmodule of โ„‹. 4. Characterization of STPNS-Aideals by special kind of submodules In this section we introduce the characterization of the notion of STPNS-Aideals by special kind of submodules. The following proposition gives characterization of the notion of STPNS-Aideals. Proposition 4.1 Let โ„‹ be a finitelygenerated Zโ€“regularmultiplication content-Rโ€“module and I is a STPNS- Aideal of R with annR( โ„‹ ) ๏ƒ I if and only if Iโ„‹ is STPNS-Asubmodule of โ„‹. Proof (โ‡’) Let ๐ต2๐ฟ ๏ƒ I โ„‹ for B is an ideal of ๐‘…and ๐ฟ โŠ† โ„‹ . But โ„‹ is a multiplicationRโ€“module , then ๐ฟ = Jโ„‹ forsome ideal J of ๐‘… , that is ๐ต2๐ฟ = ๐ต2 Jโ„‹๏ƒ I โ„‹. But โ„‹ is a finitely generatedmultiplication, then by lemma 2.18 we have ๐ต2J๏ƒ I + annR( โ„‹ ). Since annR( โ„‹ ) ๏ƒ I then I + annR( โ„‹ ) = I, that ๐ต 2j๏ƒ I. But I is STPNS-Aideal of R then either ๐ตj โŠ† I + ๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…) or ๐ต2 โˆˆ [I + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…] = I + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)). It means that either ๐ตjโ„‹ โŠ† Iโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐ต2โ„‹ โŠ† Iโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). Since โ„‹ is a Z โ€“ regular and content R-module then by lemmas 2.12, 2.14 ๐‘†๐‘œ๐‘( โ„‹ ) = ๐‘†๐‘œ๐‘( ๐‘… )โ„‹ and ๐ฝ( โ„‹) = ๐ฝ( ๐‘… )โ„‹. So that either BL ๏ƒIโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) or ๐‘ข2โ„‹ ๏ƒIโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) . Therefore by lemma 2.5 Iโ„‹ is STPNS-Asubmodule of โ„‹. (โ‡) Let ๐‘ข2J ๏ƒ ๐ผ for ๐‘ข โˆˆ ๐‘… and J is an ideal of R , it means that ๐‘ข2J โ„‹๏ƒ ๐ผโ„‹. But โ„‹ is multiplication, then ๐‘ข2๐ฝโ„‹ = ๐‘ข2๐ฟ โŠ† ๐ผโ„‹. But ๐ผโ„‹ is a STPNS-Athen by lemma 2.7 either ๐‘ข๐ฟ โŠ† ๐ผโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) or ๐‘ข2โ„‹ โŠ† ๐ผโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ). But โ„‹ is Z-Regular content Rโ€“module, then by lemmas 2.12, 2.14 ๐‘†๐‘œ๐‘( โ„‹) = ๐‘†๐‘œ๐‘( ๐‘… )โ„‹, and ๐ฝ( โ„‹ ) = ๐ฝ( ๐‘… )โ„‹. Thus either ๐‘ข๐ฝโ„‹๏ƒ๐ผโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐‘ข2โ„‹๏ƒ๐ผโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). That is either ๐‘ข๐ฝ ๏ƒ ๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐‘ข2 โˆˆ ๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) = [๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…]. Therefore by lemma 2.7 I is STPNS-Aideal of R. Proposition 4.2 Let โ„‹ be a finitelygenerated multiplicationprojective Rโ€“module and I is a STPNS-Aideal of R with annR( โ„‹ ) ๏ƒ I if and only if Iโ„‹ is STPNS-Asubmodule of โ„‹. Proof (โ‡’) Let ๐ด2โ„Ž ๏ƒ Iโ„‹ for ๐ด is a submodule of โ„‹ and h โˆˆ โ„‹. Since โ„‹ isamultiplication Rโ€“ module , then ๐ด = ๐ตโ„‹ and โ„Ž = ๐ฝโ„‹ forsome ideals ๐ต, J of ๐‘… , that is ๐ด2โ„Ž = (๐ตโ„‹)2 Jโ„‹๏ƒ I โ„‹. But โ„‹ is a finitelygenerated multiplication , then by lemma 2.18 we have ๐ต2J๏ƒ I + annR( โ„‹ ). Since annR( โ„‹ ) ๏ƒ I then I + annR( โ„‹ ) = I, that ๐ต 2J๏ƒ I. But I is STPNS-Aideal of R, then either ๐ตJ โŠ† I + ๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…) or ๐ต2 โˆˆ [I + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…] = I + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)). It means that either ๐ตjโ„‹ โŠ† Iโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐ต2โ„‹ โŠ† Iโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). Since โ„‹ is a projective Rโ€“module then by lemmas 2.9, 2.10 ๐‘†๐‘œ๐‘( โ„‹ ) = ๐‘†๐‘œ๐‘( ๐‘… )โ„‹ and ๐ฝ( โ„‹) = ๐ฝ( ๐‘… )โ„‹. So that either Ah ๏ƒIโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) or โ„Ž2 ๏ƒIโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ). Therefore by corollary 3.3 Iโ„‹ is STPNS-Asubmodule of โ„‹. IHJPAS. 36 (3) 2023 312 (โ‡) Let ๐‘ข2J ๏ƒ ๐ผ for ๐‘ข โˆˆ ๐‘… and J is an ideal of R , it means that ๐‘ข2J โ„‹๏ƒ ๐ผโ„‹ . But โ„‹ is multiplication, then ๐‘ข2๐ฝโ„‹ = ๐‘ข2๐ฟ โŠ† ๐ผโ„‹. But ๐ผโ„‹ is a STPNS-Athen by lemma 2.7 either ๐‘ข๐ฟ โŠ† ๐ผโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) or ๐‘ข2โ„‹ โŠ† ๐ผโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ).But โ„‹ is projective Rโ€“ module, then by lemmas 2.9, 2.10 ๐‘†๐‘œ๐‘( โ„‹) = ๐‘†๐‘œ๐‘( ๐‘… )โ„‹ and ๐ฝ( โ„‹ ) = ๐ฝ( ๐‘… )โ„‹. Thus either ๐‘ข๐ฝโ„‹๏ƒ๐ผโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐‘ข2โ„‹๏ƒ๐ผโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). That is either ๐‘ข๐ฝ ๏ƒ ๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐‘ข2 โˆˆ ๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) = [๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…]. Therefore by lemma 2.17 I is STPNS-Aideal of R. Proposition 4.3 Let โ„‹ be a finitelygenerated multiplicationnon-singularcontentRโ€“module and I is a STPNS- Aideal of R with annR( โ„‹ ) ๏ƒ I if and only if Iโ„‹ is STPNS-Asubmodule of โ„‹. Proof (โ‡’) Let ๐ต2๐ฟ ๏ƒ I โ„‹ for B is an ideal of ๐‘… and ๐ฟ โŠ† โ„‹ . Since โ„‹ is a multiplication Rโ€“module , then ๐ฟ = Jโ„‹ forsome ideal J of ๐‘… , that is ๐ต2๐ฟ = ๐ต2 Jโ„‹๏ƒ I โ„‹. But โ„‹ is a finitelygenerated multiplication , then by lemma 2.18 we have ๐ต2J๏ƒ I + annR( โ„‹ ). Since annR( โ„‹ ) ๏ƒ I then I + annR( โ„‹ ) = I, that ๐ต 2j๏ƒ I. But I is STPNS-Aideal of R, then either ๐ตj โŠ† I + ๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…) or ๐ต2 โˆˆ [I + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…] = I + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)). It means that either ๐ตjโ„‹ โŠ† Iโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐ต2โ„‹ โŠ† Iโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). Since โ„‹ is a non-singular content-Rโ€“module then by lemmaies 2.16, 2.14 ๐‘†๐‘œ๐‘( โ„‹ ) = ๐‘†๐‘œ๐‘( ๐‘… )โ„‹ and ๐ฝ( โ„‹) = ๐ฝ( ๐‘… )โ„‹. So that either BL ๏ƒIโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) or ๐‘ข2โ„‹ ๏ƒIโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) . Therefore by lemma 2.2 Iโ„‹ is STPNS-Asubmodule of โ„‹. (โ‡) Let ๐‘ข2v โŠ† ๐ผ for ๐‘ข, ๐‘ฃ โˆˆ ๐‘…, it means that ๐‘ข2v โ„‹๏ƒ ๐ผโ„‹ . But ๐ผโ„‹ is a STPNS-Athen by lemma 2.7 either ๐‘ข๐‘ฃโ„‹ โŠ† ๐ผโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) or ๐‘ข2 โˆˆ ๐ผโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ). But โ„‹ is a non-singular content-Rโ€“module then by lemmas 2.16, 2.14 ๐‘†๐‘œ๐‘( โ„‹ ) = ๐‘†๐‘œ๐‘( ๐‘… )โ„‹ and ๐ฝ( โ„‹) = ๐ฝ( ๐‘… )โ„‹. Thus either ๐‘ข๐‘ฃโ„‹ โŠ† ๐ผโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐‘ข2โ„‹๏ƒ๐ผโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) . That is either ๐‘ข๐‘ฃ โˆˆ ๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐‘ข2 โˆˆ ๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) = [๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…]. Therefore I is STPNS-A ideal of R. Proposition 4.4 Let โ„‹ be a faithfulfinitely generatedmultiplication Rโ€“moduleand I is a STPNS-Aideal of R with annR( โ„‹ ) ๏ƒ I if and only if Then Iโ„‹ is STPNS-Asubmodule of โ„‹. Proof (โ‡’) Let โ„Ž2๐พ ๏ƒ Iโ„‹ for h โˆˆ โ„‹ and ๐พ โŠ† โ„‹ . Since โ„‹ is a multiplicationRโ€“module , then โ„Ž = ๐ตโ„‹and ๐พ = Jโ„‹ forsome ideals ๐ต, J of ๐‘… , that is โ„Ž2๐พ = (๐ตโ„‹)2 Jโ„‹๏ƒ I โ„‹. But โ„‹ is a finitelygenerated multiplication , then by lemma 2.18 we have ๐ต2J๏ƒ I + annR( โ„‹ ). Since annR( โ„‹ ) ๏ƒ I then I + annR( โ„‹ ) = I, that ๐ต 2J๏ƒ I. But I is STPNS-Aideal of R, then either ๐ตJ โŠ† I + ๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…) or ๐ต2 โˆˆ [I + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…] = I + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)). It means that either ๐ตjโ„‹ โŠ† Iโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐ต2โ„‹ โŠ† Iโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹). Since โ„‹ is faithful then by lemmas 2.8, 2.9 ๐‘†๐‘œ๐‘( โ„‹ ) = ๐‘†๐‘œ๐‘( ๐‘… )โ„‹ and ๐ฝ( โ„‹) = ๐ฝ( ๐‘… )โ„‹. So that either hK ๏ƒIโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) or โ„Ž2 ๏ƒIโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) . Therefore by corollary 3.4 Iโ„‹ is STPNS-Asubmodule of โ„‹. (โ‡) Let ๐‘ข2J ๏ƒ ๐ผ for ๐‘ข โˆˆ ๐‘… and J is an ideal of R , it means that ๐‘ข2J โ„‹๏ƒ ๐ผโ„‹ . But โ„‹ is multiplication, then ๐‘ข2๐ฝโ„‹ = ๐‘ข2๐ฟ โŠ† ๐ผโ„‹. But ๐ผโ„‹ is a STPNS-Athen by lemma 2.7 either ๐‘ข๐ฟ โŠ† ๐ผโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ) or ๐‘ข2โ„‹ โŠ† ๐ผโ„‹ + (๐ฝ(โ„‹) โˆฉ ๐‘ ๐‘œ๐‘(โ„‹) ).But โ„‹ is faithful, then by IHJPAS. 36 (3) 2023 313 lemmas 2.5, 2.6 ๐‘†๐‘œ๐‘( โ„‹ ) = ๐‘†๐‘œ๐‘( ๐‘… )โ„‹ and ๐ฝ( โ„‹) = ๐ฝ( ๐‘… )โ„‹ . Thus either ๐‘ข๐ฝโ„‹๏ƒ๐ผโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) or ๐‘ข2โ„‹๏ƒ๐ผโ„‹ + (๐ฝ(๐‘…)โ„‹ โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)โ„‹) .That is either ๐‘ข๐ฝ ๏ƒ ๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) or ๐‘ข2 โˆˆ ๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)) = [๐ผ + (๐ฝ(๐‘…) โˆฉ ๐‘ ๐‘œ๐‘(๐‘…)):๐‘… ๐‘…]. Therefore by lemma 2.7 I is STPNS-Aideal of R. Proposition 4.5 Let โ„‹ be a faithfulfinitely generatedmultiplicationRโ€“module, and โ„ฑ is a propersubmodule of โ„‹ . Then the followingstatements are valent : 1. โ„ฑ is STPNS-Asubmodule of โ„‹. 2. [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . 3. โ„ฑ = Iโ„‹ for some STPNS-Aideal I of R . Proof ( 1 ) ๏ƒ› ( 2 ) Follows by proposition 4.4. ( 2 ) ๏ƒž ( 3 ) Suppose that [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R. But โ„‹ is a multiplication , then โ„ฑ = [โ„ฑ:๐‘… โ„‹]โ„‹. Put I = [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R and โ„ฑ = Iโ„‹. ( 3 ) ๏ƒž ( 2 ) Suppose that โ„ฑ = Iโ„‹ for some STPNS-Aideal of R. Since โ„‹ is a multiplication , โ„ฑ = [โ„ฑ:๐‘… โ„‹]โ„‹ = Iโ„‹. But โ„‹ is a faithfulfinitely generated , then by lemma 2.18 โ„‹ is cancellation , therefore [โ„ฑ:๐‘… โ„‹] = I is STPNS-Aideal of R. Proposition 4.6 Let โ„‹ be a finitelygenerated multiplicationZโ€“regular content R โ€“ module, and โ„ฑ is a propersubmodule of โ„‹ withannR (โ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]. Then the followingstatements are valent : 1. โ„ฑ is STPNS-Asubmodule of โ„‹. 2. [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . 3. โ„ฑ = Iโ„‹ for some STPNS-Aideal I of R with annR (โ„‹) โŠ† ๐ผ. Proof ( 1 ) ๏ƒ› ( 2 ) Follows byproposition 4.1. ( 2 ) ๏ƒž ( 3 ) Let โ„ฑ โŠ† โ„‹ , then โ„ฑ = [โ„ฑ:R โ„‹]โ„‹ ( for โ„‹ isamultiplication) . Put [โ„ฑ:R โ„‹] = I, impliesthat I is STPNS-Aideal of R with annR (โ„‹) = [0:R โ„‹] โŠ† [โ„ฑ:R โ„‹] = I , that is annR (โ„‹) โŠ† I. ( 3 ) ๏ƒž ( 2 ) Supposethat โ„ฑ = Iโ„‹ forsome STPNS-Aideal of R with annR (โ„‹) โŠ† I . But โ„‹ isamultiplication , then โ„ฑ = [โ„ฑ:R โ„‹]โ„‹ = Iโ„‹, and since โ„‹ is a finitelygenerated then by lemma 2.20 โ„‹ is a weakcancellation , it means that [โ„ฑ:R โ„‹] + annR (โ„‹) = I + annR (โ„‹). But annR (โ„‹) โŠ† I, it means that I + annR (โ„‹) = I and annR (โ„‹) โŠ† [โ„ฑ:R โ„‹] , it means that [โ„ฑ:R โ„‹] + annR (โ„‹) = [โ„ฑ:R โ„‹]. Thus [โ„ฑ:R โ„‹] = I, but I is STPNSโ€“2-A ideal of R, itfollows that [โ„ฑ:R โ„‹] is STPNS-Aideal of R. Proposition 4.7 Let โ„‹ be a finitelygenerated multiplicationprojective Rโ€“module, and โ„ฑ is a propersubmodule of โ„‹ with annR (โ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]. Then the followingstatements are valent : 1. โ„ฑ is STPNS-Asubmodule of โ„‹. 2. [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . 3. โ„ฑ = Iโ„‹ forsome STPNS-Aideal I of R withannR (โ„‹) โŠ† ๐ผ. Proof ( 1 ) ๏ƒ› ( 2 ) Followsby proposition 4.2. ( 2 ) ๏ƒ› ( 3 ) Followssimilarly as inproposition 4.6. IHJPAS. 36 (3) 2023 314 Proposition 4.8 Let โ„‹ be a finitelygenerated multiplicationnon-singular content Rโ€“module, and โ„ฑ is a propersubmodule of โ„‹ with annR (โ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]. Then the following statementsare valent : 1. โ„ฑ is STPNS-Asubmodule of โ„‹. 2. [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . 3. โ„ฑ = Iโ„‹ for some STPNS-Aideal I of R withannR (โ„‹) โŠ† ๐ผ. Proof ( 1 ) ๏ƒ› ( 2 ) Follows by proposition 4.3. ( 2 ) ๏ƒ› ( 3 ) Follows similarly as in proposition 4.6. 5. Conclusion We will present the most important propositions in this research: . Let โ„‹ be a faithfulfinitely generatedmultiplication Rโ€“module, and โ„ฑ is a propersubmodule of โ„‹ . Then thefollowing statementsare valent : 1. โ„ฑ is STPNS-Asubmodule of โ„‹. 2. [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . 3. โ„ฑ = Iโ„‹ for some STPNS-Aideal I of R . . Let โ„‹ be a finitelygenerated multiplicationZโ€“regular content R โ€“ module, and โ„ฑ is a propersubmodule of โ„‹ with annR (โ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]. Then the followingstatements are valent : 1. โ„ฑ is STPNS-Asubmodule of โ„‹. 2. [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . 3. โ„ฑ = Iโ„‹ for some STPNS-Aideal I of R with annR (โ„‹) โŠ† ๐ผ. . Let โ„‹ be a finitelygenerated multiplicationprojective Rโ€“module, and โ„ฑ is a propersubmodule of โ„‹ with annR (โ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]. Then the followingstatements are valent : 1. โ„ฑ is STPNS-Asubmodule of โ„‹. 2. [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . 3. โ„ฑ = Iโ„‹ for some STPNS-Aideal I of R with annR (โ„‹) โŠ† ๐ผ. . Let โ„‹ be a finitelygenerated multiplicationnon-singular content Rโ€“module, and โ„ฑ is a propersubmodule of โ„‹ with annR (โ„‹) โŠ† [โ„ฑ:๐‘… โ„‹]. Then thefollowing statements are valent : 1. โ„ฑ is STPNS-Asubmodule of โ„‹. 2. [โ„ฑ:๐‘… โ„‹] is STPNS-Aideal of R . 3. โ„ฑ = Iโ„‹ for some STPNS-Aideal I of R with annR (โ„‹) โŠ† ๐ผ. References 1. Lu, C. P. Prime Submodule of Modules, Comm. Math, University Spatula, 1981 , 33, 61-69. 2. Mohamad, E. D. Haibat, K. M. 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