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Accepted Manuscript
Journal of Algebra and its Applications
Article Title: Radical factorization for trivial extensions and amalgamated duplication
rings
Author(s): Tiberiu Dumitrescu, Najib Mahdou, Youssef Zahir
DOI: 10.1142/S0219498821500250
Received: 15 August 2018
Accepted: 26 November 2019
To be cited as: Tiberiu Dumitrescu, Najib Mahdou, Youssef Zahir, Radical factorization
for trivial extensions and amalgamated duplication rings, Journal of Al-
gebra and its Applications, doi: 10.1142/S0219498821500250
Link to final version: https://doi.org/10.1142/S0219498821500250
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RADICAL FACTORIZATION FOR TRIVIAL EXTENSIONS AND
AMALGAMATED DUPLICATION RINGS
TIBERIU DUMITRESCU, NAJIB MAHDOU, AND YOUSSEF ZAHIR
Abstract. Let A⊆Bbe a commutative ring extension such that Bis a
trivial extension of A(denoted by A∝E) or an amalgamated duplication of
Aalong some ideal of A(denoted by A ./ I). This paper examines the transfer
of AM-ring, N-ring, SSP-ring and SP-ring between Aand B. We study the
transfer of those properties to trivial ring extension. Call a special SSP-ring
an SSP-ring of the following type: it is the trivial extension of B×Cby a
C-module E, where Bis an SSP-ring, Ca von Neumann regular ring and E
a multiplication C-module. We show that every SSP-ring with finitely many
minimal primes which is a trivial extension is in fact special. Furthermore, we
study the transfer of the above properties to amalgamated duplication along
an ideal with some extra hypothesis. Our results allows us to construct non-
trivial and original examples of rings satisfying the above properties.
1. Introduction
Let Abe a commutative unitary ring (like all rings in this note). Say that an ideal
Iof Ahas radical factorization if Iis a product of radical ideals. In [21], Vaughan
and Yeagy introduced and studied SP-domains, that is, integral domains whose
ideals have radical factorization (this terminology comes from “semiprime ideal” -
another name for radical ideal). If Dis an SP-domain, then Dis almost-Dedekind
(i.e. DMis a discrete rank one valuation domain (DVR) for each maximal ideal M
of D), cf. [21, Theorem 2.4], while the converse is not true, see [11, Example 3.4.1].
For an extensive study of SP-domains see Olberding’s paper [20].
In [1], Ahmed and the first named author of this note extended SP-domain
concept to rings with zero-divisors in two different ways. A ring Ais called an SSP-
ring (resp. SP-ring) if every ideal (resp. regular ideal) has radical factorization.
Here SSP-ring is an abbreviation for “special SP-ring” and a regular ideal is an
ideal containing a regular element (i.e. not zero-divisor).
Each SSP-ring is an AM-ring, cf. [1, Theorem 3.3]. Recall that Ais an almost
multiplication ring (AM-ring) if each ideal of Awith prime radical is a prime power
(equivalently, each localization of Aat a maximal ideal is a DVR or a special
principal ideal ring (SPIR)), see [5, page 272], [15, page 16] or [16, Chapter IX]. An
SPIR (also called a special primary ring) is a local ring whose ideals are powers of
the maximal ideal Mand Mis nilpotent and principal, see [16, page 206].
Similarly, each Marot SP-ring is an N-ring, cf. [1, Theorem 2.6]. Recall that A
is an N-ring if each regular ideal of Awith prime radical is a prime power, see [15].
2000 Mathematics Subject Classification. 13D05, 13D02.
Key words and phrases. SP-ring, SSP-ring, N-ring, AM-ring, trivial extension, amalgamated
duplication of a ring along an ideal.
1
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2 TIBERIU DUMITRESCU, NAJIB MAHDOU, AND YOUSSEF ZAHIR
Also Ais a Marot ring if its regular ideals are generated by regular elements, see
[13, page 31]. As we proceed to study the above-mentioned classes of rings, the
reader may find it helpful to keep in mind the following figure with non-reversible
arrows.
SSP-ring
Marot
AM-ring SP-ring
N-ring
)
PPPPPP
Pq
HHH
Hj
Let Abe a ring and Ebe an A-module. The following ring construction called
the trivial extension of Aby E(also called the idealization of E) was introduced
by Nagata [19, page 2]. It is the ring A∝Ewhose underlying abelian group is
A×Ewith multiplication given by (a, e)(a0, e0)=(aa0, ae0+a0e). The canonical
projection A∝E→Awhose kernel 0 ∝Ehas null square induces an inclusion
preserving bijection Spec(A∝E)→Spec(A). Specifically, a prime ideal Pof A
corresponds to the prime ideal P∝Eof A∝E. Suitable background on trivial
ring extensions is provided in [2, 3, 4, 13, 14].
Let Abe a ring and Ibe an ideal of A. The following ring construction called
the amalgamated duplication of Aalong Iwas introduced by D’Anna in [8]. It is
the subring A I of A×Aconsisting of all pairs (x, y)∈A×Awith x−y∈
I. Motivations and additional applications of the amalgamated duplication are
discussed in detail in [8, 9].
Let Abe a ring and Ebe an A-module. We denote by Z(A) the set of zero-
divisors of A, by Z(E) the set of zero-divisors on E, by Reg(A) the set of regular
elements of A(i.e. Reg(A) = A\Z(A)), by Ann(E) the annihilator of E. All
rings in this note are commutative with a nonzero unit. Any unexplained fact or
terminology is standard like in [10] or [16] .
The aim of this note is to study SSP-ring, AM-ring, SP-ring and N-ring properties
for A∝E(in Section 2) and for A I (in Section 3). We can now specify more
precisely the main purposes of this paper. In section 2, we investigate the transfer
of the above properties on the trivial rings extension which completely generalize
well-known results of [1]. We prove that A∝Eis an AM-ring if and only if Ais an
AM-ring and, for each maximal ideal M∈Supp(E), AMis a field and EM'AM,
where Supp(E) denotes the support of E(Proposition 2.2). Moreover, we prove
that A∝Eis an N-ring if and only if E=ESand each ideal of Anot disjoint of S
having a prime radical is a prime power, where S=A\(Z(A)∪Z(E)) (Theorem
2.11). Section 3 deals with the study of the above properties in amalgamated
duplication of ring along an ideal and prove the following results. A I is an
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RADICAL FACTORIZATION FOR TRIVIAL EXTENSIONS AND AMALGAMATED DUPLICATION RINGS3
AM-ring if and only if Ais an AM-ring and IM= 0 for each M∈Max(A)\V(I).
As a corollary, we provide that A I is an SSP-ring (resp. AM-ring) if and only
if Ais an SSP-ring (resp. AM-ring) and I is idempotent. Finally, in the last two
of our principal results, we show under the additional hypothesis I=aI for each
a∈Reg(A) that A I is an SP-ring (resp. N-ring) if and only if Ais an SP-ring
(resp. N-ring).
2. Trivial extensions
Throughout this section, Ais a ring and Eis an A-module. We study SSP-ring,
AM-ring, SP-ring and N-ring properties for trivial extension A∝E. Let us recall
some known definitions and facts. Eis a multiplication module if each submodule
of Ehas the form IE for some ideal Iof A. Following [3], we call an ideal of A∝E
homogeneous if it has the form I∝F={(a, x)|a∈I, x ∈F}where Iis an ideal
of Aand Fis a submodule of Esuch that IE ⊆F. It can be checked directly (see
also the last paragraph of page 12 in [3]) that a product of two homogeneous ideals
is homogeneous. Our first lemma collects a few simple useful facts.
Lemma 2.1. The following assertions hold.
(a)Any SSP-ring is an AM-ring. The converse is true if the ring is local.
(b)If A∝Eis an SSP-ring, then Eis a multiplication module and every ideal
of A∝Eis homogeneous.
(c)If Ais a von Neumann regular ring and Ea multiplication module, then
A∝Eis an SSP-ring.
Proof. The direct implication in (a) is [1, Theorem 3.3] while the converse is clear
from definitions. (b) By [3, Theorem 3.2], the radical ideals of A∝Ehave the form
I∝Ewith Ia radical ideal of A. Then every ideal of A∝Eis homogeneous, as
remarked before Lemma 2.1. Since 0 ∝F(with Fsubmodule of E) is a product of
radical ideals, we get easily that F=JE for some ideal Jof A. (c) is [1, Proposition
3.6].
We get the following result, where Supp(E) denotes the support of E.
Proposition 2.2. A∝Eis an AM-ring if and only if Ais an AM-ring and, for
each maximal ideal M∈Supp(E),AMis a field and EM'AM. In particular, if
Ais local and E6= 0, then A∝Eis an AM-ring if and only if Ais a field and
E'A.
Proof. Set B=A∝E. It is known that the canonical map B→Ainduces
a bijection between Max(B) and Max(A). Moreover, the corresponding ideal of
M∈Max(A) is N=M∝Eand BNis isomorphic to AM∝EM. Therefore,
it suffices to prove the “in particular” assertion. (⇐) is clear because A∝A'
A[X]/(X2). (⇒)A∝Eis a local AM-ring with zero-divisors, hence it is an SPIR,
cf. Lemma 2.1(a). Apply [3, Lemma 4.10] to complete.
Corollary 2.3. Assume that A∝Eis an AM-ring and E6= 0. The following
assertions hold.
(a)Each M∈Supp(E)is a height zero maximal ideal.
(b)Eis flat.
(c)If Max(A)⊆Supp(E), then Ais von Neumann regular.
(d)If Ais a domain, then Ais a field and E'A.
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4 TIBERIU DUMITRESCU, NAJIB MAHDOU, AND YOUSSEF ZAHIR
Proof. (a) Let P∈Supp(E) and Mbe a maximal ideal containing P; hence M∈
Supp(E). By Proposition 2.2, AMis a field, so P=M. (b) By Proposition 2.2, E
is (locally) flat. (c) holds because all localizations of Aat maximal ideals are fields,
cf. Proposition 2.2. (d) follows from (a) and Proposition 2.2.
Corollary 2.4. When Ehas an element with zero annihilator, the following are
equivalent.
(a)A∝Eis an SSP-ring,
(b)A∝Eis an AM-ring,
(c)Ais von Neumann regular and E'A.
Proof. (a)⇒(b) follows from Lemma 2.1(a), while (c)⇒(a) follows from Lemma
2.1(c). (b)⇒(c) There exists a monomorphism h:A→E, so Supp(E) = S pec(A),
hence Ais von Neumann regular, cf. Corollary 2.3(c). By Proposition 2.2, his
(locally) an isomorphism.
Example 2.5. Let Bbe an SSP-ring, Ca von Neumann regular ring and E6= 0 a
multiplication C-module. Set A=B×Cand consider Eas an A-module by scalar
restriction via A→C. Then A∝Eis an SSP-ring because it is the direct product
of B∝0'Band the SSP-ring C∝E(cf. Lemma 2.1(c)), so [1, Proposition 3.1]
applies.
Call a ring A∝Elike in Example 2.5 a special SSP-ring (obtained from B,C,E).
As shown below, an SSP-ring of the form A∝Eis often a special one.
Proposition 2.6. Suppose that A∝Eis an SSP-ring such that E6= 0 and Ahas
finitely many minimal prime ideals. Then A∝Eis a special SSP-ring.
Proof. By [1, Corollary 3.4], A=A1× · ·· × Anwhere each Aiis an SPIR or an
SP-domain. Then A∝E= (A1∝E1)×... ×(An∝En) where Ei= (0 ×···× 0×
Ai×0× · ·· × 0)E. Relabel to get Ei= 0 for 1 ≤i≤kand Ei6= 0 for i>k. By
the final assertion of Proposition 2.2 or part (d) of Corollary 2.3, Aiis a field and
Ei'Aifor i > k. Then B=A1× · ·· × Akis an SSP-ring (a direct product of
SSP-rings), C=Ak+1 × · ·· × Anis a von Neumann regular ring (a finite product
of fields) and E'Cis a multiplication C-module. So A∝Eis isomorphic to the
special SSP-ring obtained from B,C,E.
Note that the result above applies when A∝Eis a semilocal SSP-ring. Indeed,
as an epimorphic image of A∝E,Ais a semilocal AM-ring, cf. Lemma 2.1(a).
Since the localizations of Aat its maximal ideals are DVRs or SPIRs, we see that
Ahas finitely many minimal prime ideals.
Theorem 2.7. Suppose that A∝Eis an SSP-ring and E6= 0. If Eand Ann(E)
are finitely generated, then A∝Eis a special SSP-ring.
Proof. Set I=Ann(E). As Eis finitely generated, S upp(E) = V(I). By Corollary
2.3(c) applied to the A/I-module E, we get that A/I is von Neumann regular. Pick
afrom I. By Lemma 2.1(b) and [3, Theorem 3.3 (5)], there exists g∈Asuch that
ag =aand gE =aE = 0, so g∈Iand I=I2. As Iis finitely generated, Iis
generated by some idempotent element e. Then A=B×C, where B=Ae =I
and C=A(1 −e). Clearly, BE = 0 and C'A/I is von Neumann regular. Hence
A∝Eis a special SSP-ring obtained from B,C,E.
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RADICAL FACTORIZATION FOR TRIVIAL EXTENSIONS AND AMALGAMATED DUPLICATION RINGS5
The preceding proof shows that if A∝Eis an SSP-ring and E6= 0 is finitely
generated, then A/Ann(E) is von Neumann regular and Ann(E) is idempotent.
But Ann(E) is not necessarily finitely generated, as our next example shows.
Example 2.8. Let Dbe an SP-domain which is not Dedekind, so it has a maximal
ideal Nwhich is not finitely generated (see [20]). Let x∈Nwith NDN=xDN.
By [1, Proposition 3.1], A=D/xD is an SSP-ring and we may arrange that A
is not von Neumann regular (multiply xby some element in Q2−Nwhere Qis
another maximal ideal of D). Set M=N A. It follows that M AM= 0 and one
can check locally that IM =Ifor each ideal I⊆Mof A. Set B=A∝E
where E=A/M. We claim that Bis an SSP-ring (note that AnnA(E) = Mis not
finitely generated). We show first that every ideal of Bis homogeneous by verifying
condition (5) in [3, Theorem 3.3], that is, for each a∈A, there exists some b∈A
with a=ab and bE =aE. When a /∈M, take b= 1. If a∈M, take b∈Msuch
that a=ab (such a bexists because aM =aA as noted above). Hence every ideal
of Bhas the form I∝0 or I∝Ewhere Iis an ideal of A(with I⊆Min the first
case). Let Ibe an ideal of Awith radical factorization I1· ··In. If I6⊆ M, then
(I1∝E)··· (In∝E) is a radical factorization of I∝E. Assume that I⊆M. We
may assume that I1⊆Mand Ij6⊆ Mfor j≥2.To see this, consider the inverse
image Jof Iin D, take a radical factorization of Jin D(which has at most one
factor contained in Nsince x∈J−N2) and reduce this factorization modulo xD.
We have radical factorizations I∝E= (I1∝E)···(In∝E) and I∝0 = (I1∝
E)··· (In∝E)(M∝E). So every ideal of Bhas radical factorization.
Consider the multiplicative set S=A−(Z(A)∪Z(E)) of A. Then Ecan be
identified with a submodule of ESand note that E=ESif and only if E=sE for
each s∈S. The following result is [1, Proposition 2.4].
Proposition 2.9. With Sdefined above, A∝Eis an SP-ring if and only if
E=ESand each ideal of Anot disjoint of Shas a radical factorization.
In order to give an N-ring variant of the preceding result, we need the following
lemma.
Lemma 2.10. For a surjective ring morphism p:B→A, consider the following
two assertions.
(a)Bis an N-ring.
(b)If His an ideal of A,Q=√His prime and p−1(H)is regular, then His a
power of Q.
Then (a)implies (b)and the converse is also true if each regular ideal of B
contains ker(p).
Proof. (a)⇒(b) Let Hbe an ideal of Awhich has prime radical Qand such
that p−1(H) is regular. We have pp−1(H) = p−1(√H) = p−1(Q), so p−1(H) =
(p−1(Q))nfor some n≥1, as Bis an N-ring. Thus H=p(p−1(H)) = p((p−1(Q))n) =
Qnas desired.
We prove that (b)⇒(a) under additional hypothesis that each regular ideal of B
contains ker(p). Let Jbe a regular ideal of Bwith prime radical Q. Set H=p(J).
By our assumption, Q⊇J⊇ker(p), so J=p−1(H). As √H=p(√J) = p(Q), we
get p(J) = p(Q)nfor some n≥1, by (b). Thus J=Qnbecause Qnis regular, so
it contains ker(p).
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6 TIBERIU DUMITRESCU, NAJIB MAHDOU, AND YOUSSEF ZAHIR
Recall that a Pr¨ufer ring is a ring whose finitely generated regular ideals are
invertible, see [12].
Theorem 2.11. With S=A−(Z(A)∪Z(E)),A∝Eis an N-ring if and only
if E=ESand each ideal of Anot disjoint of Shaving a prime radical is a prime
power.
Proof. Set B=A∝Eand let p:B→Abe the canonical map. Assume that
Bis an N-ring. Then Bis a Pr¨ufer ring, so E=ES(cf. [15, Theorem 3] and [3,
Theorem 4.16(2)]), hence each regular ideal of Bcontains ker(p)=0∝E, cf. [3,
Theorem 3.9]. By [3, Theorem 3.5], Reg(B) = S∝E. So, for an ideal Hof A,
p−1(H) = H∝Eis regular if and only if His not disjoint of S. Now Theorem
2.11 clearly follows from Lemma 2.10.
Corollary 2.12. Assume that Ais a domain and Eis torsion-free. Then A∝E
is an N-ring if and only Ais an almost Dedekind domain and Eis divisible.
Proof. Apply Theorem 2.11 with S=A−{0}(note that Eis divisible if and only
if E=ES).
3. Amalgamated duplication rings
Throughout this section, Ais a ring and Ian ideal of A. We study SSP-ring,
AM-ring, SP-ring and N-ring properties for amalgamated duplication ring A I .
We begin with the following AM-ring result connected to [6, Corollary 3.8].
Theorem 3.1. A I is an AM-ring if and only if Ais an AM-ring and IM= 0
for each M∈Max(A)∩V(I). In particular, if A I is an AM-ring, then Iis
idempotent.
Proof. Set B=A I. (⇒) As Bis an AM-ring, so is its epimorphic image A.
Let M∈Max(A)∩V(I) and let Nbe the unique prime ideal of Blying over M
(we embed Adiagonally in B). By [8, Proposition 7(a)], BN'AM IM, hence
we may assume that Bis local ring. As Bis an AM-ring, it is a DVR or an SPIR.
Then I= 0 because, if x∈I− {0}, then the ideals (x, 0)Band (0, x)Bare not
comparable under inclusion. (⇐) Since being an AM-ring is a local property, it
suffices to show that Band Ahave the same localizations at the maximal ideals.
Let Nbe a maximal ideal of Band Mits contraction to A. If I⊆M, then
BN'AM IM=AMsince IM= 0 by our assumption. If I*M, then
BN'AM. Finally, the “in particular” assertion follows from the main one because
an ideal whose localizations are zero or the whole ring is idempotent.
Corollary 3.2. Assume that Iis finitely generated. Then A I is an SSP-
ring (resp. AM-ring) if and only if Ais an SSP-ring (resp. AM-ring) and Iis
idempotent.
Proof. (⇒) Since A I is an SSP-ring (resp. AM-ring), so is its epimorphic
image A, (cf. [1, Proposition 3.1(a)] for SSP-rings while AM-ring case is clear).
By Theorem 3.1, Iis idempotent (remember that every SSP-ring is an AM-ring).
(⇐) As Iis idempotent and finitely generated, Iis generated by some idempotent
element e. Then, Ais ring-isomorphic to direct product I×Cwhere C=A(1 −e).
So A I '(I×C) (I×0) 'I×I×Cis an SSP-ring (resp. AM-ring), since
being an SSP-ring or AM-ring are properties which are inherited by factors and
finite direct products.
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RADICAL FACTORIZATION FOR TRIVIAL EXTENSIONS AND AMALGAMATED DUPLICATION RINGS7
We shall give our SP-ring result for A I. The next lemma prepares the way.
Lemma 3.3. For a surjective ring morphism p:B→A, consider the following
two assertions.
(a)Bis an SP-ring.
(b)If His an ideal of Asuch that p−1(H)is regular, then Hhas a radical
factorization.
Then (a)implies (b)and the converse is also true if each regular ideal of B
contains ker(p).
Proof. (a)⇒(b) Let Hbe an ideal of Asuch that p−1(H) is regular. Since
Bis an SP-ring, p−1(H) has a radical factorization p−1(H) = J1···Jn. Then
H=p(p−1(H)) has radical factorization p(J1)·· · p(Jn), where p(Ji) is radical
since Ji⊇ker(p).
We prove that (b)⇒(a) under additional hypothesis that each regular ideal of
Bcontains ker(p). Let Jbe a regular ideal of B, so J⊇ker(p). Set H=p(J).
Since p−1(H) = Jis regular, Hhas a radical factorization H=H1··· Hn. Then
p−1(Hi) is a regular radical ideal for each i. So K:= p−1(H1)···p−1(Hn) is a
regular ideal, hence it contains ker(p), by our assumption. Since p(K) = p(J), we
get K=J, so Jhas a radical factorization.
Here is the promised SP-ring result.
Theorem 3.4. Let Abe a ring and Ibe an ideal of A. Consider the following two
assertions.
(a)A I is an SP-ring,
(b)Ais an SP-ring.
Then (a)implies (b)and the converse is also true if I=aI for each a∈Reg(A).
Proof. Set B=A I and consider the surjective ring morphism p:B→A
given by p(x, y) = x. By [17, Proposition 2.2], Reg(B) = (Reg(A)×Reg(A)) ∩B.
(a)⇒(b) If His a regular ideal of A, then p−1(H) is regular (if x∈Reg(A)∩H,
then (x, x)∈p−1(H)). Apply Lemma 3.3.
We prove that (b)⇒(a) under additional hypothesis that I=aI for each
a∈Reg(A). Let (b, a)∈Reg(B) and i∈I. Since I=aI,i=aj for some j∈I.
We get (0, i)=(b, a)(0, j ), so (b, a)B⊇0×I=ker(p). Thus each regular ideal of
Bcontains ker(p). Apply Lemma 3.3.
Remark 3.5. (a) If A I is a Marot SP-ring and Ais local, then I=aI for each
a∈Reg(A) (combine [1, Theorem 2.6], [15, Theorem 3] and [6, Theorem 2.2]).
(b) If I2= 0, then A I 'A∝I, so A I is an SP-ring if and only if A
is an SP-ring and I=aI for each a∈Reg(A), cf. Proposition 2.9. As a specific
example, take A=Z∝Qand I= 0 ∝Z. By [1, Proposition 2.4], Ais an SP-ring,
while A I is not.
(c) Probably the next case that could be completely settled is when Iconsists of
nilpotent elements, since, in this case, every radical ideal of A I contains 0 I, so
it has the form K I for some radical ideal Kof A, because (A I )/(0 I)'A.
We close by giving an N-ring variant of the preceding result.
Theorem 3.6. Let Abe a ring and Ibe an ideal of A. Consider the following two
assertions.
(a)A I is an N-ring,
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8 TIBERIU DUMITRESCU, NAJIB MAHDOU, AND YOUSSEF ZAHIR
(b)Ais an N-ring.
Then (a)implies (b)and the converse is also true if I=aI for each a∈Reg(A).
Proof. The proof is similar to that of Theorem 3.4 using Lemma 2.10 instead of
Lemma 3.3.
ACKNOWLEDGMENTS. The authors would like to express their sincere
thanks to the anonymous referee for his/her insightful suggestions toward the im-
provement of the paper.
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(2007), 321-343.
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Tiberiu Dumitrescu, Department of Mathematics, University of Bucharest, Romania.
E-mail address: tiberiu@fmi.unibu.ro, tiberiu dumitrescu2003@yahoo.com
Najib Mahdou, Laboratory of Modeling and Mathematical Structures, Department
of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M.
Ben Abdellah Fez, Morocco. E-mail address: mahdou@hotmail.com
ZAHIR Youssef, Laboratory of Modeling and Mathematical Structures, Department
of Mathematics, Faculty of Science and Technology of Fez, Box 2202, University S.M.
Ben Abdellah Fez, Morocco. E-mail address: youssef.zahir@usmba.ac.ma
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