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Corrigendum to “On multiplication (m,n) -hypermodules” [European J. Combin. 44 (2015) 153–171]
European Journal of Combinatorics 44 (2015) 172–174
Contents lists available at ScienceDirect
European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc
Corrigendum
Corrigendum to ‘‘On multiplication (m, n)-hypermodules’’ [European J. Combin. 44 (2015) 153–171] R. Ameri a , M. Norouzi b a
School of Mathematics, Statistic and Computer Sciences, University of Tehran, Tehran, Iran
b
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran
article
info
Article history: Available online 29 November 2014
In [1], we need to define some new notations and assumptions, that without using them, some parts of the proof of some theorems and corollaries are incorrect. Therefore, in this corrigendum, we introduce these notations and assumptions and also consider related corrections. Moreover, some minor missprints are corrected. The authors would like to apologize for any inconvenience caused. 1. In Definition 2.4, ‘‘(iv) {0} = g (r1i−1 , 0, rin+−11 , x)’’. 2. In Definition 2.5, ‘‘g (R(n−1) , N ) ∈ P ∗ (N )’’. (n−2)
, m) ⊆ g (R, 1(Rn−2) , m)’’. m m 4. Page 6, line 4, ‘‘f ( ) = h( ) = i=1 xi = i=1 xi =’’. 5. Page 7, Lemma 3.7 is not true (is removed) and XP (M ) is not a subhypermodule of M. Hence, we define the set XP (M ) accompany with XP (M ) and consider some hypotheses as follows: (n−2) XP (M ) = m ∈ M | ∃p ∈ P ; 0 ∈ g h(1R , −p, 0(m−2) ), 1R ,m (n−2) XP (M ) = m ∈ M | ∃s ∈ P ; {0} = g h(1R , −s, 0(m−2) ), 1R ,m .
3. Page 5, line 1, ‘‘0 ∈ g (0, 1R xm 1
xm 1
(n−2)
(n−2)
Also, we let x ∈ g (A, 1R , m) implies that there exists b ∈ A such that {x} = g (b, 1R and {0} = h(r , −r , 0(m−2) ), for all x, m ∈ M and r , A ⊆ R. By attention to case (5), we reconstruct the results in [1], as follows: 6. Theorem 3.8 in [1] is changed to the following new theorems:
DOI of original article: http://dx.doi.org/10.1016/j.ejc.2014.08.002. E-mail addresses: [email protected] (R. Ameri), [email protected] (M. Norouzi). http://dx.doi.org/10.1016/j.ejc.2014.11.005 0195-6698/© 2014 Elsevier Ltd. All rights reserved.
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R. Ameri, M. Norouzi / European Journal of Combinatorics 44 (2015) 172–174
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Theorem 3.7. If (M , f , g ) is a multiplication (m, n)-hypermodule over (R, h, k), then for every maximal hyperideal P of R either M = XP (M ) or M is n-ary P-cyclic. Proof. The ‘‘if’’ part of the proof of Theorem 3.8 in [1]. Theorem 3.8. Let (M , f , g ) be an (m, n)-hypermodule over (R, h, k) such that for every maximal hyperideal P of R either M = XP (M ) or M is n-ary P-cyclic. Then M is multiplication. Proof. The ‘‘converse’’ part of the proof of Theorem 3.8 in [1] with the following minor changes: (i) ‘‘M = XP (M ) or M is n-ary P-cyclic’’ changed to ‘‘M = XP (M ) or M is n-ary P-cyclic’’. (n−2) (ii) ‘‘If M = XP (M ), then y ∈ XP (M ) and so 0 ∈ g h(1R , −s, 0(m−2) ), 1R , y ’’ changed to ‘‘If (n−2)
M = XQ (M ), then y ∈ XQ (M ) and so g h(1R , −s, 0(m−2) ), 1R
, y = {0}’’. (iii) ‘‘h(1R , −s, 0(m−2) ) ⊆ K ⊆ Q ’’ changed to ‘‘h(1R , −s, 0(m−2) ) ⊆ L ⊆ Q ’’.
7. In the statement of Corollary 3.9, Corollary 3.10, Theorem 3.16 and Corollary 3.19 in [1], we have to add the hypothesis ‘‘XP (M ) = XP (M ) for every maximal hyperideal P of R’’ (also, XI (M ) = XI (M ) in Corollary 3.10). 8. In the proof of Corollary 3.9 in [1] (i) lines 4 and 5 replaced by ‘‘Then, for q ∈ P we have h(1R , −q, 0(m−2) ) ⊆ Iα , and so (n−2) , M ⊆ g (Iα , 1(Rn−2) , M ) = g (R, 1(Rn−2) , mα ),’’. g h(1R , −q, 0(m−2) ), 1R (ii) line 9 replaced by ‘‘M (n−2)
= f(k) g (R, 1(Rn−2) , mλ )(λ∈Λ) ⊆ f(k) g (P , 1R(n−2) , M )(λ∈Λ) ⊆
g (P , 1R , M ) ⊆ M’’. (iii) lines 16 and 17 replaced by ‘‘By the proof of Theorem 3.7 (in corrigendum), it follows that mλ ∈ XP (M ). Hence M = XP (M ) = XP (M ) (by supposition). Therefore, (M , f , g ) is a multiplication (m, n)-hypermodule by Theorem 3.8 (in corrigendum). (n−2)
9. In proof of Corollary 3.10 in [1], ‘‘r ∈ XP (I )’’, ‘‘0 ∈ k h(1R , −t , 0(m−2) ), 1R
, r ’’, ‘‘0 ∈ (n−2) , m) ⊆’’, ‘‘XP g (I , 1(Rn−2) , M ) ’’, ‘‘0 ∈ g h(1R , −s, 0(m−2) ), 1R(n−2) , m ’’ and ‘‘0 = g (0, 1R (n−2) g (r , 1R , 0) ∈’’ changed to ‘‘r ∈ XP (I )’’, ‘‘{0} = k h(1R , −t , 0(m−2) ), 1R(n−2) , r ’’, ‘‘{0} = (n−2) , m) =’’, ‘‘XP g (I , 1(Rn−2) , M ) ’’, ‘‘{0} = g h(1R , −s, 0(m−2) ), 1R(n−2) , m ’’ and ‘‘{0} = g ({0}, 1R (n−2) g (r , 1R , {0}) =’’, respectively.
10. In proof of Theorem 3.11 in [1], (n−2) , x . Hence, for t ∈ h(1R , −p, 0(m−2) ) (i) line 15–20 replaced by ‘‘0 ∈ g h(1R , −p, 0(m−2) ), 1R (n−2)
we have 0 ∈ g (t , 1R , x). Since M is faithful, it implies that 0 ∈ h(1R , −p, 0(m−2) ) and so 1R ∈ P which is a contradiction.’’ (ii) page 12, line 15–20 replaced by ‘‘
{0} = g (1(Rn−1) , 0) ⊆ g 1(Rn−1) , g h(aτ , −aλ , 0(m−2) ), 1(Rn−2) , m = g k h(aτ , −aλ , 0(m−2) ), 1(Rn−1) , 1(Rn−2) , m . Since M is faithful, we have 0 ∈ h(aτ , −aλ , 0(m−2) ) which implies aτ = aλ , because (R, h) is canonical.’’ 11. In the statement of Theorem 3.15 in [1], R is a Krasner (m, n)-hyperrings and I is a normal hyperideal, and also the following minor changes are applied in the proof: (i) ‘‘h(1R , I , 0(m−2) ) ⊆’’ and ‘‘h(1R , I , 0(m−2) ) ∈’’ changed to ‘‘h(1R , I , 0(m−2) ) =’’. (ii) ‘‘(m, n)-field’’ changed to ‘‘(m, n)-hyperfield’’. (iii) Page 14, line 1–5 replaced by ‘‘Hence, since (R, h) is canonical and L is a hyperideal, then 1R ∈ h(1, 0(m−1) ) ⊆ h(1R , I , 0(m−2) )
= h k(r , a, 1(Rn−2) ), I , 0(m−2) ⊆ h(L, L, 0(m−2) ) ⊆ L.’’
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12. Line 2–4 in the proof of Lemma 3.14 in [1] replaced by ‘‘by [2], we have (n−2)
G r , 1R
, f (m, N , 0(m−2) ) = f (t , N , 0(m−2) ) | t ∈ g (r , 1R(n−2) , m) ⊆ {f (t , N , 0(m−2) ) | t ∈ N } = {N } = 0M /N .’’
13. In the proof of Theorem 3.16 in [1] (n−2) (i) ‘‘0 ∈ g (r , 1R , m)’’ changed to ‘‘{0} = g (r , 1(Rn−2) , m)’’. (ii) ‘‘Then h(1R , −p, 0(m−2) ) ⊆ I ⊆ P, and so 1R ∈ P which is a contradiction.’’ replaced by ‘‘Since (n−2) , m) XP (M ) = XP (M ), then, there exists q ∈ P such that {0} = g (h(1R , −q, 0(m−2) ), 1R (n−2)
, m). Thus h(1R , −q, 0(m−2) ) ⊆ and so for all t ∈ h(1R , −q, 0(m−2) ) we have {0} = g (t , 1R (m−2) I ⊆ P. This implies that 1R ∈ h(q, P , 0 ) ⊆ P, which is a contradiction.’’ (iii) ‘‘vector space over (m, n)-field’’ changed to ‘‘(m, n)-hypervector space over (m, n)hyperfield’’. 14. In the proof of Lemma 4.4 in [1] (i) ‘‘s ∈ R’’ and ‘‘p ∈ P’’ changed to ‘‘S ⊆ R’’ and ‘‘P1 ⊆ P’’. (n−2) (ii) ‘‘and so k(a, s, 1R ) = p. Hence’’ changed to ‘‘and so k(a, s, 1R(n−2) ) ∈ P1 ⊆ P for some s ∈ S. Hence’’. (n−2) ⊆ P’’ changed to ‘‘⊆ k h(1R , −q, 0(m−2) ), P , 1(Rn−2) ⊆ P’’. (iii) ‘‘= k h(1R , −q, 0(m−2) ), p, 1R References
[1] R. Ameri, M. Norouzi, On multiplication (m, n)-hypermodules, European J. Combin. 44 (2015) 153–171. [2] R. Ameri, M. Norouzi, V. Leoreanu, On prime and primary subhypermodules of (m, n)-hypermodules, European J. Combin. 44 (2015) 175–190.
Contents lists available at ScienceDirect
European Journal of Combinatorics journal homepage: www.elsevier.com/locate/ejc
Corrigendum
Corrigendum to ‘‘On multiplication (m, n)-hypermodules’’ [European J. Combin. 44 (2015) 153–171] R. Ameri a , M. Norouzi b a
School of Mathematics, Statistic and Computer Sciences, University of Tehran, Tehran, Iran
b
Department of Mathematics, Faculty of Basic Sciences, University of Bojnord, Bojnord, Iran
article
info
Article history: Available online 29 November 2014
In [1], we need to define some new notations and assumptions, that without using them, some parts of the proof of some theorems and corollaries are incorrect. Therefore, in this corrigendum, we introduce these notations and assumptions and also consider related corrections. Moreover, some minor missprints are corrected. The authors would like to apologize for any inconvenience caused. 1. In Definition 2.4, ‘‘(iv) {0} = g (r1i−1 , 0, rin+−11 , x)’’. 2. In Definition 2.5, ‘‘g (R(n−1) , N ) ∈ P ∗ (N )’’. (n−2)
, m) ⊆ g (R, 1(Rn−2) , m)’’. m m 4. Page 6, line 4, ‘‘f ( ) = h( ) = i=1 xi = i=1 xi =’’. 5. Page 7, Lemma 3.7 is not true (is removed) and XP (M ) is not a subhypermodule of M. Hence, we define the set XP (M ) accompany with XP (M ) and consider some hypotheses as follows: (n−2) XP (M ) = m ∈ M | ∃p ∈ P ; 0 ∈ g h(1R , −p, 0(m−2) ), 1R ,m (n−2) XP (M ) = m ∈ M | ∃s ∈ P ; {0} = g h(1R , −s, 0(m−2) ), 1R ,m .
3. Page 5, line 1, ‘‘0 ∈ g (0, 1R xm 1
xm 1
(n−2)
(n−2)
Also, we let x ∈ g (A, 1R , m) implies that there exists b ∈ A such that {x} = g (b, 1R and {0} = h(r , −r , 0(m−2) ), for all x, m ∈ M and r , A ⊆ R. By attention to case (5), we reconstruct the results in [1], as follows: 6. Theorem 3.8 in [1] is changed to the following new theorems:
DOI of original article: http://dx.doi.org/10.1016/j.ejc.2014.08.002. E-mail addresses: [email protected] (R. Ameri), [email protected] (M. Norouzi). http://dx.doi.org/10.1016/j.ejc.2014.11.005 0195-6698/© 2014 Elsevier Ltd. All rights reserved.
, m),
R. Ameri, M. Norouzi / European Journal of Combinatorics 44 (2015) 172–174
173
Theorem 3.7. If (M , f , g ) is a multiplication (m, n)-hypermodule over (R, h, k), then for every maximal hyperideal P of R either M = XP (M ) or M is n-ary P-cyclic. Proof. The ‘‘if’’ part of the proof of Theorem 3.8 in [1]. Theorem 3.8. Let (M , f , g ) be an (m, n)-hypermodule over (R, h, k) such that for every maximal hyperideal P of R either M = XP (M ) or M is n-ary P-cyclic. Then M is multiplication. Proof. The ‘‘converse’’ part of the proof of Theorem 3.8 in [1] with the following minor changes: (i) ‘‘M = XP (M ) or M is n-ary P-cyclic’’ changed to ‘‘M = XP (M ) or M is n-ary P-cyclic’’. (n−2) (ii) ‘‘If M = XP (M ), then y ∈ XP (M ) and so 0 ∈ g h(1R , −s, 0(m−2) ), 1R , y ’’ changed to ‘‘If (n−2)
M = XQ (M ), then y ∈ XQ (M ) and so g h(1R , −s, 0(m−2) ), 1R
, y = {0}’’. (iii) ‘‘h(1R , −s, 0(m−2) ) ⊆ K ⊆ Q ’’ changed to ‘‘h(1R , −s, 0(m−2) ) ⊆ L ⊆ Q ’’.
7. In the statement of Corollary 3.9, Corollary 3.10, Theorem 3.16 and Corollary 3.19 in [1], we have to add the hypothesis ‘‘XP (M ) = XP (M ) for every maximal hyperideal P of R’’ (also, XI (M ) = XI (M ) in Corollary 3.10). 8. In the proof of Corollary 3.9 in [1] (i) lines 4 and 5 replaced by ‘‘Then, for q ∈ P we have h(1R , −q, 0(m−2) ) ⊆ Iα , and so (n−2) , M ⊆ g (Iα , 1(Rn−2) , M ) = g (R, 1(Rn−2) , mα ),’’. g h(1R , −q, 0(m−2) ), 1R (ii) line 9 replaced by ‘‘M (n−2)
= f(k) g (R, 1(Rn−2) , mλ )(λ∈Λ) ⊆ f(k) g (P , 1R(n−2) , M )(λ∈Λ) ⊆
g (P , 1R , M ) ⊆ M’’. (iii) lines 16 and 17 replaced by ‘‘By the proof of Theorem 3.7 (in corrigendum), it follows that mλ ∈ XP (M ). Hence M = XP (M ) = XP (M ) (by supposition). Therefore, (M , f , g ) is a multiplication (m, n)-hypermodule by Theorem 3.8 (in corrigendum). (n−2)
9. In proof of Corollary 3.10 in [1], ‘‘r ∈ XP (I )’’, ‘‘0 ∈ k h(1R , −t , 0(m−2) ), 1R
, r ’’, ‘‘0 ∈ (n−2) , m) ⊆’’, ‘‘XP g (I , 1(Rn−2) , M ) ’’, ‘‘0 ∈ g h(1R , −s, 0(m−2) ), 1R(n−2) , m ’’ and ‘‘0 = g (0, 1R (n−2) g (r , 1R , 0) ∈’’ changed to ‘‘r ∈ XP (I )’’, ‘‘{0} = k h(1R , −t , 0(m−2) ), 1R(n−2) , r ’’, ‘‘{0} = (n−2) , m) =’’, ‘‘XP g (I , 1(Rn−2) , M ) ’’, ‘‘{0} = g h(1R , −s, 0(m−2) ), 1R(n−2) , m ’’ and ‘‘{0} = g ({0}, 1R (n−2) g (r , 1R , {0}) =’’, respectively.
10. In proof of Theorem 3.11 in [1], (n−2) , x . Hence, for t ∈ h(1R , −p, 0(m−2) ) (i) line 15–20 replaced by ‘‘0 ∈ g h(1R , −p, 0(m−2) ), 1R (n−2)
we have 0 ∈ g (t , 1R , x). Since M is faithful, it implies that 0 ∈ h(1R , −p, 0(m−2) ) and so 1R ∈ P which is a contradiction.’’ (ii) page 12, line 15–20 replaced by ‘‘
{0} = g (1(Rn−1) , 0) ⊆ g 1(Rn−1) , g h(aτ , −aλ , 0(m−2) ), 1(Rn−2) , m = g k h(aτ , −aλ , 0(m−2) ), 1(Rn−1) , 1(Rn−2) , m . Since M is faithful, we have 0 ∈ h(aτ , −aλ , 0(m−2) ) which implies aτ = aλ , because (R, h) is canonical.’’ 11. In the statement of Theorem 3.15 in [1], R is a Krasner (m, n)-hyperrings and I is a normal hyperideal, and also the following minor changes are applied in the proof: (i) ‘‘h(1R , I , 0(m−2) ) ⊆’’ and ‘‘h(1R , I , 0(m−2) ) ∈’’ changed to ‘‘h(1R , I , 0(m−2) ) =’’. (ii) ‘‘(m, n)-field’’ changed to ‘‘(m, n)-hyperfield’’. (iii) Page 14, line 1–5 replaced by ‘‘Hence, since (R, h) is canonical and L is a hyperideal, then 1R ∈ h(1, 0(m−1) ) ⊆ h(1R , I , 0(m−2) )
= h k(r , a, 1(Rn−2) ), I , 0(m−2) ⊆ h(L, L, 0(m−2) ) ⊆ L.’’
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12. Line 2–4 in the proof of Lemma 3.14 in [1] replaced by ‘‘by [2], we have (n−2)
G r , 1R
, f (m, N , 0(m−2) ) = f (t , N , 0(m−2) ) | t ∈ g (r , 1R(n−2) , m) ⊆ {f (t , N , 0(m−2) ) | t ∈ N } = {N } = 0M /N .’’
13. In the proof of Theorem 3.16 in [1] (n−2) (i) ‘‘0 ∈ g (r , 1R , m)’’ changed to ‘‘{0} = g (r , 1(Rn−2) , m)’’. (ii) ‘‘Then h(1R , −p, 0(m−2) ) ⊆ I ⊆ P, and so 1R ∈ P which is a contradiction.’’ replaced by ‘‘Since (n−2) , m) XP (M ) = XP (M ), then, there exists q ∈ P such that {0} = g (h(1R , −q, 0(m−2) ), 1R (n−2)
, m). Thus h(1R , −q, 0(m−2) ) ⊆ and so for all t ∈ h(1R , −q, 0(m−2) ) we have {0} = g (t , 1R (m−2) I ⊆ P. This implies that 1R ∈ h(q, P , 0 ) ⊆ P, which is a contradiction.’’ (iii) ‘‘vector space over (m, n)-field’’ changed to ‘‘(m, n)-hypervector space over (m, n)hyperfield’’. 14. In the proof of Lemma 4.4 in [1] (i) ‘‘s ∈ R’’ and ‘‘p ∈ P’’ changed to ‘‘S ⊆ R’’ and ‘‘P1 ⊆ P’’. (n−2) (ii) ‘‘and so k(a, s, 1R ) = p. Hence’’ changed to ‘‘and so k(a, s, 1R(n−2) ) ∈ P1 ⊆ P for some s ∈ S. Hence’’. (n−2) ⊆ P’’ changed to ‘‘⊆ k h(1R , −q, 0(m−2) ), P , 1(Rn−2) ⊆ P’’. (iii) ‘‘= k h(1R , −q, 0(m−2) ), p, 1R References
[1] R. Ameri, M. Norouzi, On multiplication (m, n)-hypermodules, European J. Combin. 44 (2015) 153–171. [2] R. Ameri, M. Norouzi, V. Leoreanu, On prime and primary subhypermodules of (m, n)-hypermodules, European J. Combin. 44 (2015) 175–190.