Microsoft Word - 118-125 Mathematics | 118 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2001 Vol. 31 (3) 2018 WN-2-Absorbing Submodules and WNS-2-Absorbing Submodules Wissam A. Hussain Departmentt of"Mathematics ,College of Education for Pure science, Tikrit"University,Iraq Wissam.abbas1987@gmail.com 'Haibt K. Mohammd ali 'Department of"Mathematics,College"of'"Computer 'Science"and"Mathematics, Tikrit"University,Iraq'/ ///' Article history: Received 30 July 2018, Accepted 3 September 2018, Published December 2018"" Abstract"" "In''this"article, we"study, the"concept"of WN"-"2"-''Absorbing'''submodules and WNS''- ''2''-''Absorbing"submodules as generalization of weakly 2-absorbing and weakly semi 2- absorbing submodules respectively. We investigate some of basic properties, examples and characterizations of them. Also, prove, the class of WN-2-Absorbing "submodules is contained in the class of WNS-2-Absorbing "submodules. Moreover, many interesting results about these concepts, were proven. Keywords:"WN-2-Absorbing submodules, WNS-2-Absorbing submodules, Weakly 2- Absorbing submodules, Weakly Semi-2-Absorbing submodules. 1. Introduction" Weakly''2''-''absorbing'''submodules'''was"introduced by"Darani and''"Soheilinia, in 2011, where a''proper''submodule''B of an"R''-''module"Y is''called'''weakly'''"2- absorbing''submodule,''if''whenever'''0 ≠ aby ∊ B, with a, b ∊ R, y ∊ Y, implies that either ay ∊ B or by ∊ B or ab ∊ [B:Y] [1]". And the concept of a weakly semi 2-absorbing submodule was introduce by"Haibt and Khalaf in 2018, where a''proper ''submodule''B of an"R"- '''module"'''Y'''is'''called a"weakly"semi'''2-"absorbing"submodule , if"whenever 0 ≠ a y ∊ B , with a ∊ R, y ∊ Y, implies that either ay ∊ B or'''a ∊ [ B : Y] [2]. "These two concepts are generalized in this article, to WN-2-Absorbing submodules and WNS-2-Absorbing submodules, we prove that the class of WN-2-Absorbing submodules is contained in the class of WNS-2-Absorbing submodules while the converse is not true see example (3.14).''''Recall'''that''a''submodule A of an R"-''module Y is ''called'''small if"for any"submodule B of Y, Y = A + B, implies that A = Y [3]. Recall that an R-epimorphism f ∶ Y → Y is called small if Kerf is a small submodule of Y, and f ȷ M ȷ M` ȷ f M and ȷ M f ȷ M [3]. A ring R is a good ring if 𝚥(R) Y = 𝚥(Y), where Y is an R- module equivalently R is a good ring if 𝚥(Y) ∩ A = (A) for every submodule A of Y [3]. If Y is an R-module and A, B, C are submodules of Y with B ⊆ C. Then (A + B) ∩ C = ( A ∩ C ) + (B ∩ C ) = ( A ∩ C ) + B [3]. Recall that an "R-"module Y is '''regular ''if'''R'/'ann'(x)''is''regular''ring'''[4].'''Recall''that''a''subset''S'''of'''a''ring'R is called multiplica- tively'''''closed'''subset''''of R if '''1'∈'S''and'''ab ∈ S "for all'''a'',''b'∈ S [5]". This note consists of two parts in the first part, we introduced'''the''concept''of "WN''-2"-''Absorbing''submodule'', and in the second part we introduced"the''concept''of'''WNS'-'2'-'Absorbing''''submodule'. Mathematics | 119 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2001 Vol. 31 (3) 2018 2. WN-2-Absorbing"Submodules"and Related Concept In this part of the research,'we''''introduce''''and'''studied''''the''''concept'''of WN-2"- 'Absorbing'''submodules'''as''a''generalization'''of weakly"2'-'absorbing''submodules''. 'Definition'1 A"proper'''submodule''B of an"R'-'module'''Y is''said''to"be WN-'2'-'Absorbing submodules'"if''whenever' 0 aby ∊ "B, where a, b ∊"R, y ∊ Y, implies that either ay ∊ "B ′ "ȷ Y or by ∊ "B ′ "ȷ Y or ab ∊ "B ′ "ȷ Y : Y , where 𝚥(Y) is the Jacobsen radical of Y. An''ideal' I"of a ring R' is"'said' to'"be WN-'2-'Absorbing' ideal of"R, if'''I is'''a WN-"2- 'Absorbing'''submodules'"of"an"R'-'module'R. Remark"2 'Every"weakly 2-absorbing"submodule of an R-module Y is WN-2-Absorbing submodules, while the converse is not true. Proof Clear. For the converse consider the following example : let Y = Z , R = Z and B = 〈8〉 it is clear that B is a WN-2-Absorbing submodules of Y since B + 𝚥(Y) =〈8〉 + 〈2〉 = 〈2〉. But B is not weakly 2-absorbing submodule of Y since, 0 ≠ 2.2.2 ∊ B, but 2.2 ∉ B and 2.2 ∉ [B:Y] = 8Z. 'Proposition' 3 Let''Y'''be'''an"R''-''module',''and'''B''a'''proper'''submodule'''of"Y''''with'''𝚥''(Y)''⊆B then'''B'''is'''a''''weakly"2''-'''absorbing''''submodule''of'''Y'''if''and''only"if''''B'''is"a"WN-2"- 'Absorbing'''submodule'''of'''Y. Proof" ⟹ By remark (2.2). (⟸ since 𝚥 (Y) ⊆ B then B + 𝚥 (Y) = B , hence proof is direct. Proposition 4 Let Y be an"R-module, and B a"proper"submodule of Y with A ⊂ B. If A is a WN-2- Absorbing"submodule of Y and 𝚥 (Y) ⊆𝚥(B), then A"is'''a"WN-2-''Absorbing' 'submodule"of B. 'Proof" 'Let 0" aby ∊ ′A, where′a, b ∊ R, y ∊ ′B , since A is a"WN-2-Absorbing "submodule of Y then either ay ∊ A ȷ Y or by ∊ A ȷ Y or ab ∊ A ȷ Y : Y , but 𝚥 (Y) ⊆𝚥( B), so either ay ∊ A ȷ B or by ∊ A ȷ B or ab ∊ A ȷ Y : Y ⊆ A ȷ B : Y ⊆ A ȷ B : B since B is a"submodule of Y. Hence A a WN-2-Absorbing"submodule of B. Proposition 5 Let"Y be an"R"-'module,''and'''B a'proper'''submodule''of Y,"'if B + 𝚥(Y) is''a WN-2"- Absorbing'''submodule'''of Y,''then'''B is'''a WN-2"-'Absorbing"'submodule''of'''Y. 'Proof" Since'''B ⊆ B + 𝚥(Y), hence proof is clearly. Mathematics | 120 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2001 Vol. 31 (3) 2018 Remark 6 The"intersection of two is a "WN-2"-'Absorbing''submodules'"of"an"R'-"module Y need not to be is a WN-2-Absorbing"submodule. The following example explain that: Let Y = Z, R = Z, A = 6Z, B = 7Z. Clearly A, B is a WN-2-Absorbing submodules since they are weakly 2-absorbing submodules of Y but A ∩ B = 42Z is not WN-2-Absorbing submodule of Y since, if 0 ≠ 2.3.7 ∈ A ∩ B, but 2.7 ∉ A ∩ 𝚥(Y) and 3.7 ∉ A ∩ 𝚥(Y) and 2.3 ∉ [A ∩ 𝚥(Y): Y] = 42Z. 'Proposition'7 'Let Y"be an"R'-"module ,"and''A, B are WN-'2'-'Absorbing''submodules"of''Y with A ⊆ 𝚥( Y) and B ⊆ 𝚥(Y), then A ∩ B is WN-2-Absorbing submodules of Y. Proof" Let 0 ≠ aby ∈ A ∩ B , with a, b ∈ R, y ∈ Y, implies that 0 ≠ aby ∈ A and 0 ≠ aby ∈ B. it follows that either ay ∊ "′A ′"ȷ Y or by ∊ "′A′ "ȷ Y or ab ∊ A ȷ Y : Y , and either ay ∊ ′B′ ′ȷ Y or by ∊ ′B ′ȷ Y or ab ∊ "′B ′ȷ Y : Y . But"A ⊆ 𝚥( Y) and B ⊆ 𝚥(Y), then A + 𝚥(Y) = 𝚥(Y) and B + 𝚥(Y) = 𝚥(Y). Hence ay ∊ ȷ Y or by ∊ ′ȷ Y or' ab′ ∊ ′ ȷ Y : Y . Thus'A ∩ B ⊆ 𝚥(Y), implies that A ∩ B + 𝚥(Y) = 𝚥(Y) thus, we have ay ∊ A ∩ B ȷ Y or by ∊ A ∩ B ȷ Y or ab ∊ A ∩ B ȷ Y : Y . So, A ∩ B is a WN-2- 'Absorbing'''submodule of'"Y. 'Proposition"8 ' Let"Y be an R'-"module, over"a good ring and A, B are submodules of Y, A ⊈ B and 𝚥(Y) ⊆ A , if B is WN-2-Absorbing submodules of Y, then A ∩ B is WN-2-Absorbing submodules of A. Proof Since A ⊈ B, then A ∩ B is a proper submodule of A, let 0 ≠ aby ∈ A ∩ B , with a, b ∈ R, y ∈ Y. then 0 ≠ aby ∈ A and 0 ≠ aby ∈ B. Since B WN-2-Absorbing submodules of Y, then either ay ∊ ′B′ ′ȷ Y or by ∊ ′B" "ȷ Y or ab ∊ ′B ′ȷ Y : Y . That is either either ay ∊ B" "ȷ Y ∩ A or by ∊ B "ȷ Y " ∩ A or abY ⊆ B ȷ Y ∩ A , hence by moduler law we have either ay ∊ A ∩ B ȷ A or by ∊ A ∩ B ȷ A or ab ∊ A ∩ B ȷ A : Y ⊆ A ∩ B ȷ A : A , thus A ∩ B is WN-2-Absorbing submodules of A. As a direct consequence of proposition 2.8, we get'''the''following"'corollary 'Corollary"9 'Let"Y be'''an"R'-"module", over a"good ring and A, B are"submodules of Y, A ⊈ B and A is a"maximal"submodule of Y, if B is WN-2-Absorbing"submodules of Y, then A ∩ B is WN-2-Absorbing submodules of A. Proposition 10 Let Y'''be''an"R'-'module'',"and''A proper"submodule'"of Y.''Then"A'''is"WN-2"- 'Absorbing'''submodules'''of"Y if'''and'''only'''if'''for each submodule B of Y with [ A:Y] ⊆ [A:B] and for each a, b ∊ R with 0 ≠ abB ⊆ A, implies that either aB⊆A + 𝚥(Y) or bB⊆A + 𝚥(Y) or ab ∊ [A+𝚥(Y):Y]. Mathematics | 121 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2001 Vol. 31 (3) 2018 Proof Suppose that 0 ≠ abB ⊆ A for each submodule B of Y and a, b ∊ R. then 0 ≠ aby ∊ A for each y ∊ B ⊆ Y. But A is WN-2-Absorbing submodules of Y, implies that either ay ∊ A + 𝚥(Y) or by ∊ A + 𝚥(Y) or ab ∊ [A+𝚥(Y):Y]. It follows that either aB⊆A + 𝚥(Y) or bB⊆A + 𝚥(Y) or ab ∊ [A+𝚥(Y):Y]. Conversely: let 0 ≠ aby ∊ A for all y ∊ Y, a, b ∊ R . That is 0 ≠ abY ⊆ A, implies that ab ∊ [ A:Y] ⊆ [A:B], it follows that 0 ≠ abB ⊆ A hence by hypothesis either aB⊆ A + 𝚥(Y) or bB⊆ A + 𝚥(Y) or ab ∊ [A+𝚥(Y):Y]. That is either ay ∊ A + 𝚥(Y) or by ∊ A + 𝚥(Y) or ab ∊ [A+𝚥(Y):Y]. Thus A is WN-2-Absorbing submodules of Y. 'Proposition 11 Let Y be an"R'-'module'''and"A is'''a'''proper'''submodule''of Y.'If"A''is"WN-2"-'Absorbing submodules"'of"Y, 'then" S A is"WN-2"-'Absorbing"submodules"'of an ' S R -"module" S Y , where S is''a'''multiplicatively'''closed'"subset"'of 'R. Proof Let 0 ≠ ∈ S A , where , ∊ S R and ∈ S Y with r , r ∈ R , s , s , s ∈ S , y ∊ Y. Then 0 ≠ ∈ S A , where t = s s s ∈ S , then there exists t ∈ S such that 0 ≠ t r r y ∈ A. But A is WN-2-Absorbing submodules of Y, then either t r y ∈ A + 𝚥(Y) or t r y ∈ A + 𝚥(Y) or t r r ∈ A + 𝚥(Y) : Y]. implies that ∈ S A ȷ Y ⊆ S A ȷ S Y or ∈ S A ȷ Y ⊆ S A ȷ S Y or ∈ S A + 𝚥(Y) : Y] ⊆ [ S A ȷ S Y : S Y . Thus either ∈ S A ȷ S Y or ∈ S A ȷ S Y or ∈ [S A ȷ S Y : S Y . Hence S A is WN-2-Absorbing submodules of an S R- module S Y. Proposition 12 Let h : Y → Y` be a small R-epimorphism . and A is WN-2-Absorbing submodules of Y containing Kerh. , then h A is WN-2-Absorbing submodules of of Y`. 'Proof " 'It''is''clear''that ℎ(A) is'''a'''proper'''submodule''of Y` , let aby`∈ h (A) , where a , b ∈ R, y` ∈ Y` , then h (y) = y`. for some y ∊ Y. thus 0 ≠ abh(y) ∊ h(A), then h(aby) = h(n) for some non- zero n ∊ A. since Kerh ⊆ A it follows that 0 ≠ aby ∊ A , but A is WN-2-Absorbing submodules of Y, then either ay ∊ A + 𝚥(Y) or by ∊ A + 𝚥(Y) or ab ∊ [ A + 𝚥(Y) :Y]. Thus either ah(y) ∊ h(A) +h( 𝚥(Y)) or bh(y) ∊ h(A) + h(𝚥(Y)) or abh(y) ⊆ h(A) +h( 𝚥(Y)). But h is small epimorphism then either ay`∈ h (A) + (Y`) or by`∈ h (A) + (Y`) or abY` ⊆ h (A) + 𝚥(Y`). Hence h A is WN-2-Absorbing submodules of of Y`. Proposition 13 Let h : Y → Y` be a small R-epimorphism . and A is WN-2-Absorbing submodules of Y` then ℎ 𝐴 is WN-2-Absorbing submodules of Y. Proof Let 0 ≠ aby ∊ ℎ 𝐴 , where a , b ∈ R, y ∊ Y, with ay ∉ ℎ 𝐴 + 𝚥(Y)and by ∉ ℎ 𝐴 + 𝚥(Y). It follows that ah(y) ∉ h( ℎ 𝐴 + 𝚥(Y)) = A + 𝚥(Y`) and bh(y) ∉ h( ℎ 𝐴 + 𝚥(Y)) = A Mathematics | 122 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2001 Vol. 31 (3) 2018 + 𝚥(Y`) because h is a small epimorphism. We have 0 ≠ aby ∊ ℎ 𝐴 , implies that 0 ≠ abh(y) ∊ 𝐴 , but A is WN-2-Absorbing submodules of Y`, then ab ∊ [A + 𝚥(Y`): Y`] that is ab Y` ⊆ A + 𝚥( Y` ), implies that abh(Y) ⊆ A + 𝚥( Y` ), hence abY ⊆ ℎ 𝐴 𝚥 𝑌` ⊆ ℎ 𝐴 𝚥 𝑌 . Thus ab ∊ [ℎ 𝐴 𝚥 𝑌 :Y]. 3. WNS-2-Absorbing Submodules and Related Concept This section devoted to'''introduce'''and'''study''the'''concept'''of WNS"-2- "Absorbing'''submodules''''as"a'''generalization'''of'''a''weakly"semi"2-'absorbing'''submodule. Definition 14 A''proper'''submodule'''B'''of"an"R'-'module"Y"is'"said''to'be"a"WNS'-'2'- 'Absorbing'''submodule'''of''Y , 'if"whenever' 0 ≠ 𝑎 𝑦 ∈ "𝐵 , where a ∊"R, y ∊Y, implies that either ay ∊ B + 𝚥(Y) or 𝑎 ∈ [ B + 𝚥(Y) :Y]. An"'ideal I'"of"a"'ring R"'is'''called'''a'''WNS;-"2- 'Absorbing'''ideal if"I is"a WNS-'2-'Absorbing'R-'submodule'''of an''R'-'module"R. Remarks''and'Examples 15 1. It is clear that'''every'''weakly'''semi'''2-'absorbing"submodule"'of an"R-'module"Y is'a WNS-"2-'Absorbing"submodule of Y while'''the'''converse' is'''not'''true 2. In the Z-'module" 𝑍 , the submodule B = 〈8〉 is'a WNS-2'-"Absorbing submodule'"'of Y, 'but"'not''weakly"'semi 2-absorbing of Y since 0 ≠ 2 2 ∈ 𝐵 , but 2 ∉ 𝐵 and 2 ∉ [B:Y]. 3. If Y be an R-module, with 𝚥(Y) = 0, then a WNS-2-Absorbing submodule of Y, equivalent with a weakly semi 2-absorbing submodule of Y. 4. If Y is semi simple (regular) R-module, then a WNS-2-Absorbing submodule of Y and weakly semi 2-absorbing submodule of Y are equivalent. 5. If Y is a R-"module", and B"a proper"submodule"of Y, with 𝚥(Y) ⊆ B. Then B is"a WNS-2"-Absorbing"submodule"of Y if"and"only"if B is a"weakly semi 2- "absorbing"submodule"of Y. 6. If B is a proper submodule of Y, with B + 𝚥(Y) is a WNS-2"-'Absorbing 'submodule'"of''Y, then''B is'"a WNS-"2"-'Absorbing"'submodule"of' Y. 'Proposition 16 Let Y be"an"R'-'module'''and B be'a''proper''submodule'"of'Y Then B + 𝚥(Y)'is a'"WNS-2'- "Absorbing"'submodule"of' Y 'if"and"only"if"for each non-zero a ∊ R [B + 𝚥(Y) : 𝑎 𝑦] = [B + 𝚥(Y) : ay] or 𝑎 ∊ [B + 𝚥(Y) : Y]. Proof ⟹ Suppose that 𝑎 ∉ [B + 𝚥(Y) : Y], and let c ∊ [B + 𝚥(Y) : 𝑎 𝑦], implies that 0 ≠ 𝑎 𝑐𝑦 ∊ B + 𝚥(Y), but B + 𝚥(Y) is a WNS-2-Absorbing submodule of Y and 𝑎 ∉ [B + 𝚥(Y) : Y], then acy ∊ B + 𝚥(Y), implies that c ∊ [B + 𝚥(Y) : ay]. Thus [B + 𝚥(Y) : 𝑎 𝑦] ⊆ [B + 𝚥(Y) : ay]. Clearly [B + 𝚥(Y) : ay] ⊆ [B + 𝚥(Y) : 𝑎 𝑦]. Hence [B + 𝚥(Y) : 𝑎 𝑦] = [B + 𝚥(Y) : ay]. ⟸) let 0 ≠ 𝑎 𝑦 ∊ B + 𝚥(Y), where a ∈ R, y ∈ Y. By hypothesis, if [B + 𝚥(Y) : 𝑎 𝑦] = [B + 𝚥(Y) : ay] and 0 ≠ 𝑎 𝑦 ∊ B + 𝚥(Y), implies that [B + 𝚥(Y) : 𝑎 𝑦] = R implies that [B + 𝚥(Y) : ay] = R, hence ay ∈ B +'𝚥(Y)'. Mathematics | 123 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2001 Vol. 31 (3) 2018 Proposition 17 'Let"Y 'be"'an"R'-'module"'and"A',"B are submodules of Y, with A is a subset of B. If A is a WNS-2-"Absorbing"submodule"of Y and 𝚥(Y) ⊆ (B), then A is a WNS-2-Absorbing submodule of B. Proof Similarly as in proposition 2.4 Proposition 18 Let Y be an"R'-'module"over"a good"ring''R" and"A', B are"proper'"submodules of"Y. If A is a WNS-"2-'Absorbing'"submodule'"'of"Y"then' A is' a; WNS-'2-'Absorbing''submodule''of B. 'Proof"" 'Let"0 ≠ b y ∊ B, for b ∈ R, y ∈ B ⊆ Y, it follows that either by ∈ A + 𝚥(Y) or b ∊ [A + 𝚥(Y):Y], implies that by ∈ (A + 𝚥(Y))∩ B or b y ∊ (A + 𝚥(Y)) ∩ B, for each y ∈ B. Thus by modular law, by ∈ (A∩B) + (𝚥(Y)∩ B). But R is a good ring, then 𝚥(Y)∩ B = 𝚥(B) and''A ∩'B'is'''a'''proper'''subset'of"A, hence either by ∈ 'A"+"𝚥('B") or b y ∊ A + (B) for each y ∈ B. Thus either by ∈ A + (B) or b y ∊ [A + (B):B]. Hence A is a WNS-2-Absorbing submodule of B. 'Remark 19 The intersection of two WNS-2"-"Absorbing' submodules 'of 'an"R-"'module Y"is' not necessary WNS-"2-"Absorbing'"submodules"of Y. The following example explain that: Let Y = Z, R = Z and A=2Z, B=25Z, clearly A, B are WNS-2-Absorbing submodules of Y, but A ∩ B = 50Z, is not WNS-2-Absorbing submodule of Y. 'Proposition 20 Let Y be"'an"R'-"module'","and'"A', B are"proper' submodules"of Y with 𝚥(Y) ⊆ A, or 𝚥(Y) ⊆ B, if A and B are WNS"-2-'Absorbing'"submodules"of'"Y',"then' A ∩ B]is a''WNS"-2- "Absorbing'"submodule'"of;;Y. Proof"" Let '0" ≠ r y ∈ A ∩ B, where r ∈ R, y ∈ Y, then 0 ≠ r y ∈ A and 0 ≠ r y ∈ B, but both A and B are WNS-2-Absorbing submodules of Y then either ry ∈ A + 𝚥(Y) or r ∈ A ȷ Y : Y and either ry ∈ B + 𝚥(Y) or r ∈ B ȷ Y : Y . Implies that ry ∈ A + 𝚥(Y) ∩ B + 𝚥(Y) or r Y ⊆ A ȷ Y ∩ B ȷ Y if 𝚥(Y) ⊆ B, then B + 𝚥(Y) = B , thus either ry ∈ A + 𝚥(Y) ∩ B or r Y ⊆ A ȷ Y ∩ B, it follows that either ry ∈ A∩ B + 𝚥(Y) or r Y ⊆ A ∩ B ȷ Y . that is ry ∈ A∩ B + 𝚥(Y) or r ∈ A ∩ B ȷ Y : Y . Hence A ∩ B is a WNS-2-Absorbing submodule of Y. similarly if 𝚥(Y) ⊆ A, we get A ∩ B is a WNS-2- Absorbing submodule of Y. Proposition"21" Let"Y be' an"R-"module' ,"and'"A' is a"proper'"submodule'"of' Y with 𝚥(Y) ⊆ A ,"'if A'"is' a'"WNS-2-"Absorbing' submodules'"of Y, then'"['A :Y] is"a WNS-2-"Absorbing''"ideal' of'''R. Mathematics | 124 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2001 Vol. 31 (3) 2018 Proof Since A is a WNS-2-Absorbing submodules of Y, and 𝚥(Y) ⊆ A, then by remarks and examples 15 (5), A a weakly semi 2-absorbing submodule of Y, then by [2,prop. 4], [ A:Y] is a weakly semi 2-"absorbing'"ideal' of'"R, so by (1) ['A':Y ] is"a WNS-'2-"Absorbing'"ideal' of'R'. Proposition 22 Let Y 'be"'a cyclic R'-"module ,'and"A is"a'"proper'"submodule'"of' Y if'"[A:Y]"'is a'"WNS"-2-"Absorbing'"ideal' 'of"R' with' 𝚥(R) ⊆ ['A:Y], then"A"is a WNS-2- "Absorbing'"submodules'"of'' Y. Proof Follows by remarks and examples 15(5)(1) and corollary [2, coro. 2.5]. Proposition 23 Let g : Y → Y` be small R-epimorphism and A proper submodules of Y, with Kerg ⊆ A. If A is a WNS-'2-"Absorbing'''submodules"of'"Y',"then'"g(A) is'"a' WNS-"2-"Absorbing'' submodules 'of "Y`. Proof"" Similarly,"'as"'in"'proposition'"2.12. Proposition 24 'Let′ 𝑔 ∶ "𝑌′ → ′𝑌` ""be small"R'-epimorphism and 𝐴` proper submodules of 𝑌` . If 𝐴` is a WNS-2"-"Absorbing'' submodules 'of"' 𝑌′` ,"then' 𝑔 𝐴` 'is"a''WNS-"2-"Absorbing'' submodules"of'"Y. Proof""" Similarly,"'as"'in"'proposition'"2'.13. Proposition"25 'Let"Y"be"an"R'-"module ,"and' A is'"a'"proper'"submodule'"of'' Y"if'' A is"a WNS-2- "Absorbing'"submodules"'of Y,"then' S A "is' is a WNS'-2-Absorbing' submodules of S R module S Y. Proof Similarly as in proposition 2'.11 Proposition 26 'Let"Y"be"an"R'-"module"and'"A"is"a'"WN-2-"Absorbing'"submodules"'of"Y,then'"A is'"a' WNS-"2-"Absorbing'' submodules'"of'"Y. Proof" Let"0' ≠ a y ∈ A , where a ∈ R , y ∈ Y that is 0 ≠ a. a y ∈ A . Since A is a WN-2- Absorbing. Then either ay ∈ A + 𝚥(Y) or a ∈ A ȷ Y : Y . Thus A is WNS-2-Absorbing submodules of Y. 'The"'converse"'of"'proposition '3.13 is"'not"'true. 'In"'general 'as"'the"'following"examples' shows' that: Mathematics | 125 Ibn Al-Haitham Jour. for Pure & Appl. Sci. IHJPAS https://doi.org/10.30526/31.3.2001 Vol. 31 (3) 2018 Example"27 let Y = Z ⊕ Z , R = R , A = 15Z ⊕ (0), A is WNS-2-Absorbing submodules of Y, but not is WN-2-Absorbing. Since 0 ≠ 3.5(1,0) ∊ A, but 3(1,0) ∉ A + 𝚥(Y) and 5(1,0) ∉ A + 𝚥(Y) and 3.5 ∉ [A + 𝚥(Y) : Y] = (0). References 1. Darani'. A."Y.'; Soheilnia, F. 2'-"Absorbing"and"Weakly'"2-"Absorbing' "submodules'.Taحhi Journal"Math."2011, 9',"577' – 584. 2. Haibat. M. K.; Khalaf, A.H. Weakly Semi 2-absorbing submodules. Journal of AL- Anbar University for pure Science. In press 2018. 3. Kasch, F. Modules and Rings. London Math. Soc. Monographs. 17, New York, Academic press. 1982. 4. Yaseen, M."S. F-'Regular''Modules'."M. Sc".'Thesis''university"of' Baghdad'. 1993. 5. Larsen, D. M; Mc Carthy, G. P. Multiplication Theory of Ideals. Academic Press New-York and London. 1971.