EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 3, 2024, 1855-1868 ISSN 1307-5543 – ejpam.com Published by New York Business Global One Resolution of the Conjecture between the Projective Dimension of a Simple Module S of an Artinian Ring on Zero Radical’s Cube and the First Bifunctor Extension on S Mounir Laaraj1,, Seddik Abdelalim1, Ilias Elmouki1,∗ 1 Laboratory of Fundamental and Applied Mathematics (LMFA), Faculty of Sciences Ain Chock (FSAC), University Hassan II of Casablanca (UNIVH2C), Morocco. Abstract. Through concepts from non-commutative algebra and homology, this paper resolves the conjecture that lets the first bifunctor extension to be zero when the projective dimension is finite, for a simple module S of an Artinian ring whose cube of its Jacobson radical is zero and under the condition that any simple module over this ring of finite projective dimension has a radical square zero of the cover projective of its first syzygy. For that, we use a property of the simple module which realizes the minimum of the finite projective dimensions of simple modules. Our main result is presented in the form of a corollary in the case of an Artinian ring with radical cubed zero such that the projective cover of its radical is of Loewy length two and its supremum being finite. In the part of discussions, we succeed to show the no loop conjecture through two examples. The first one is about its weak version by taking a quiver algebra A verifying J3 = 0 and without considering that rad2(P (Ω(S))) = 0 for every simple module, while the second one shows that if the extension quiver has a loop in a simple module then its projective dimension is infinite for every nilpotence index of the Jacobson radical. More importantly, we finally provide a practical third example for our special case. 2020 Mathematics Subject Classifications: 16D10, 16D20, 16D25, 16D60, 16D70, 16E05, 16E10, 16E30, 16G10, 16G20 Key Words and Phrases: Artin algebras, Representation-finite algebras, Projective dimension, Global dimension, injective dimension, Jacobson radical. 1. Introduction 1.1. Background from non-commutative algebra and homology Among the main branches of abstract algebra that deal with algebraic structures and operations are the commutative and non-commutative algebra. Both represent active re- search areas with many open problems and ongoing developments [1–3, 7, 8, 12, 16]. The ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i3.5224 Email addresses: mounirlaaraj2@gmail.com (M. Laaraj), seddikabd@hotmail.com (S. Abdelalim), i.elmouki@gmail.com (I. Elmouki) https://www.ejpam.com 1855 © 2024 EJPAM All rights reserved. M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1856 primary distinction between them is that, in commutative algebra, the order of operations is irrelevant, and ideals on left and right are identical. As for non-commutative rings, they are distinguished by the need to consider ideals on the right and left separately. It is com- mon for the study of non-commutative rings to impose a condition on one of these types of ideals without requiring that it has to be valid for its opposite side, whereas moving to the quotient of the non-commutative ring by an ideal imposes that it must be bilateral, a case which is verified by the Jacobson radical in an Artinian ring and which has several prop- erties including that of its nilpotence. Thus, the study of simple, projective and injective modules of finite type on this ring is described by idempotent primitives [10]. In parallel, homological algebra has been categorically interested in the study of the properties of the special functor Hom(M,−) and its precision, the functor Hom is only exact on the left, while the tensor product functor is only exact on the right. To obtain an exact sequence from a functor of the form Hom(M,−) (with M a module), it is possible to use an ap- proximation of M by projective modules. Such an approximation is called the projective resolution of M , and if we apply the functor Hom to a projective resolution of M , we therefore obtain a sequence which is generally not exact, but that is complex. The notion of homology corrects the lack of inexactness of this complex by associating it with a long exact sequence, called a homology sequence [10]. . This makes it possible to associate new functors with the functor Hom, called extension functors and denoted Extn(M,−) see [13]. Let A be an Artinian ring with J as its Jacobson radical, and let mod(A) be the cat- egory of left A-modules of finite type. An invariant important of A is its global dimension denoted gdim(A) and which is the supremum of projective dimensions of left A-modules of finite type, see Page 92-[13]. It is known that gdim(A) is the supremum of the projective dimensions of simple A-modules and the number of non-isomorphic simple A-modules is finite and also J is nilpotent [10]. We define therafter the quiver (that is to say a directed graph) of extensions of A, whose set of vertices is a complete set of representatives of isomorphism classes of simple A-modules, and given two vertices S and T , there is an arrow from S to T if the extension group Ext1A(S, T ) is not zero. 1.2. History of the conjecture until our contribution The first version of the conjecture that we are dealing with here, was stating that if gdim(A) is finite, then the quiver of extensions of A, is characterized by the Ext1A(S, S) null for every simple A-module S, then the second version of the conjecture, let us say the more localized and complicated, and that interests us more since the first one was resolved in the case of the supremum by Green et al. in 1985 [14]. In fact, this one was presented as the seventh conjecture in the book of representation theory of Artin Algebras [10], by the implication Ext1A(S, S) ̸= 0 ⇒ pdA(S) = ∞ which is the same as saying that if the projective dimension of each simple A-module S is finite, there is no arrow from S to S. Historically, this conjecture was demonstrated in several cases long before it was for- mally stated. At the end of 1960, Helmut Lenzing proved this conjecture in the case M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1857 where A is an algebra over an algebraically closed field [6] as it took back up the idea of Hattori-Stallings [4, 11] on the notion of (the trace of the endomorphism of a projective module). In chronological order, the next result in favor of the conjecture goes back to 1983 when Green et al. showed that conjecture is true when the global dimension of algebra is bounded by two [14]. Their proof consists of a recurrence of the number of isomorphic classes of simple modules. Subsequently, in 1986, Fuller and Zimmermann- Huisgen demonstrated in [17], the conjecture when (rad(A))3 = 0 and when A is a left serial algebra. Their approach uses the matrix Cartan filtered by radical algebra to prove that the determinant of the Cartan matrix is one. 1.1 Remark. We note that Fuller and Zimmermann-Huisgen have demonstrated that conjecture using a strong condition and which is about working with the global dimension. Our approach is interesting in the sense that we succeed to weaken that condition by proving that we can just take the projective dimension in every simple module. Later, K. Igusa proved in 1990 [5] the conjecture in a case that all algebras of endo- morphism of simple modules are separable. The author used concepts from the K-theory in his proof to point out his result included that of H. Lenzing since all fields are sepa- rable algebras. Then, the Lenzing trace function has been localized to endomorphisms of modules in mod(A) with the e-bounded projective resolution, where e is an idempotent in A and they have proved the conjecture for Artinian rings A with J2 = 0 for finite dimensional algebras over an algebraically closed field. In short, our research work aims to establish that last conjecture for Artinian rings A with J3 = 0 in the particular case where each simple module having the finite projec- tive dimension whose the projective cover of its first syzygy is canceled by J2, and this is by taking inspiration from the algebra of endomorphisms of a projective A-module as well as the Jacobson radical of this one and the characterization of simple and projective End(P )-modules with P is a projective A- module. 2. Main Theorem Given a module M in mod(A), we denote by, • Ω(M) the first syzygy, • pdM the projective dimension of M , • Ω2(M) = Ω(Ω(M)), • pd(Ω(M)) the projective dimension of the projective cover of the first syzygy of M . The following Theorem represents the main result of this paper, and in order to prove it, we will need the results that we have developed in Theorem 4.3. thereafter. 2.1 Theorem. Let A be an Artinian ring with J3 = 0. If rad2(P (Ω(S))) = 0 for every simple module S, then Ext1A(S, S) = 0. M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1858 We also fix a complete set {e1, . . . , en} of orthogonal primitive idempotents in A and let Si = Aei/Jei the simple A-module associated with ei. For convenience, we quote the following well-known result. 2.2 Lemma. Let A be an Artinian ring with a short exact sequence 0 // L //M // N // 0 in modA. The following statement holds. (1) pdN ≤ max{pdM, pdL+ 1}, and the equality occurs in case pdM ̸= pdL. (2) pdL ≤ max{pdM, pdN − 1}, and the equality occurs in case pdM ̸= pdN . (3) pdM ≤ max{pdL,pdN}, and the equality occurs in case pdN ̸= pdL+ 1. 3. Minimal projective dimension 3.1 Lemma. Let A be an Artinian ring with a radical cubed zero. Let M be a mod- ule in modA of finite projective dimension with pdM ≤ min{pd(S1), . . . ,pd(Sn)} and rad2(M) = 0. If f : P → M is a projective cover of M , then rad(ΩM) = rad2(P ). Proof. Let f : P → M be a projective cover of M . Then, ΩM ⊆ rad(P ), and hence, rad(ΩM) ⊆ rad2(P ). Since rad2(M) = 0, rad2(P ) ⊆ ΩM . Suppose that rad2(P ) ⊈ rad(ΩM). Then there exists a maximal submodule L of ΩM such that rad2(P ) ⊈ L. Since rad3(A) = 0, rad2(P ) is semi-simple. Thus, S ⊈ L where S is some simple submodule of rad2(P ), and consequently, ΩM = S ⊕ L. This yields that pd(S) ≤ pd(ΩM) < pdM ≤ pd(S), a contradiction. The proof is completed. 3.2 Lemma. Let A be an Artinian ring with radical cubed zero, and let S be the simple module of minimal projective dimension among the simple modules in modA. If pdS < ∞ and rad2(P1) = 0 with P1 is the projective cover of Ω(S), then pdS ≤ 1. Proof. Suppose that S admits a minimal projective resolution 0 // Pm // · · · // P2 // P1 // P0 // S // 0, where m > 1 then, Ω2(S) ̸= 0 and Since pd(Ω(S)) < pd(S) and like A has radical cubed zero we have, rad2(Ω(S)) = rad2(Je) = J3e = 0 where e is the idempotent associated with S, then by Lemma 1.1 and according to the hypothesis of our present Lemma, rad(Ω2(S)) = rad2(P1) = 0, and therefore Ω2(S) is a semi-simple module, then pd(S) ≤ pd(Ω2(S)) which is absurd because pd(Ω2(S)) = pd(S)− 2. Recall that if M and N are two modules, by choosing a projective resolution P∗ of M , then ExtnA(M,N) = Hn(HomA(P∗, N)) is the nth co-homology of the cochain complex of M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1859 K-modules HomA(P∗, N) which is ... −→ 0 −→ 0 −→ HomA(P0, N) −→ HomA(P1, N) −→ HomA(P2, N) −→ ... where HomA(Pn, N) is in degree n and P∗ the complex deduced from the projective reso- lution of M , see [10] . 3.3 Theorem. Let A be an Artinian ring and S = Ae/Je be a simple A-module with e primitive idempotent such that pd(S) ≤ 1, then Ext1A(S, S) = 0. Proof. if S is projective ie pd(S) = 0, then the result holds . If pd(S) = 1 and Ext1A(S, S) ̸= 0, then Hom(Je, S) ̸= 0, therefore Je is projective then Ae is isomorphic to a summand of Je since Ae is a projective cover of S, therefore l(Ae) ⩽ l(Je) and since l(Je) ⩽ l(Ae), then l(S) = 0 and S = 0 contradiction. 4. Jacobson radical of end(p) with p an A-projective module. Let P be an A-projective module in modA, we consider the algebra E = EndA(P ) and we denote J(E) the Jacobson radical of the algebra EndA(P ) or otherwise J(E) = radEndA(P )(EndA(P )) and according to Lemma 1 in [15], we have, J(E) = {Φ ∈ E/Φ(P ) is a small submodule of P} 4.1 Proposition. For n ≥ 0 and P an A-projective module in modA we have: (1) Jn(E) ⊆ Hom(P, radn A(P )) (2) radn End(P )(Hom(P,M)) ⊆ Hom(P, radn A(M)) Proof. For the first assertion, if n = 0 is obvious, we have equality. If Φ ∈ J(E), we still have by Lemma 1-[15], Φ(P ) ⊆ radA(P ), because otherwise, there exists a maximal submodule m of P such that Φ(P ) +m = P but m ̸= P and this con- tradicts that Φ(P ) is small in P , and then consequently Φ ∈ Hom(P, radA(P )) and the inclusion is true for n = 1. Similarly, we have, J2(E) = { ∑ g ◦ f/f ∈ J(E) et g ∈ J(E)}, then f(P ) ⊆ radA(P ), and g(f(P )) ⊆ g(radA(P )), that is to say g ◦ f(P ) ⊆ g(radA(A)P ), and thus g ◦ f(P ) ⊆ radA(A)g(P ) and g ◦ f(P ) ⊆ rad2 A(A)P because g(P ) ⊆ radA(P ), hence J2(E) ⊆ Hom(P, rad2 A(P )), so the Proposition is true by induction, while the sec- ond inclusion comes from the fact that Hom(P,M) is End(P )-module on the right and radEnd(P )(Hom(P,M)) = Hom(P,M).J(E) 5. Conjecture for Artinian ring with J3 = 0 Let A be an Artinian ring whose J3 = 0 and let {Pi = Aei}1≤i≤n the complete set of non-isomorphic indecomposable projective A-modules, for any simple A-module S we as- sume that rad2(P (Ω(S))) = 0 with P (Ω(S)) being the projective cover of the first syzygy M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1860 Ω(S), let I be the set such that any simple module Si for i ∈ I has a finite projective dimension, let I1 ⊆ I such that top(Pj) ≃ Si1 for j ∈ I1 where Si1 is the simple A-module of minimal projective dimension among the simple modules in modA for all i ∈ I1 . We consider the algebra Γ1 = End( ⊕ i∈I−I1 Pi) op and we denote Hom( ⊕ i∈I−I1 Pi, S) by S̃ for every simple A-module S, then we have the following Proposition. 5.1 Proposition. The ring Γ1 is Artinian with: (1) rad3 Γ1 (Γ1) = 0. (2) pd(Hom( ⊕ i∈I−I1 Pi, S)) < ∞ for every simple A-module S such that pd(S) < ∞. (3) rad2P (Ω(S̃)) = 0 with P (Ω(S̃)) the projective cover of the first syzygy Ω(S̃). Proof. It is easy to verify that Γ1 is an Artinian ring, we have by Proposition 2.1, J3(Γ1) ⊆ Hom(Γ1, rad 3 A( ⊕ i∈I−I1 Pi)), as J3 = 0, then rad3 Γ1 (Γ1) = 0, and since there is an equivalence between the two categories mod(A) and mod(Γ1) see Proposition 2.5 in [10], then the projective Γ1-modules are of the form Hom( ⊕ i∈I−I1 Pi, Q) denoted by Q̃ with Q is a projective A-module and the simple Γ1-modules are of the form Hom( ⊕ i∈I−I1 Pi, S) with S a simple A-module, the same if pd(S) = m for a simple A-module S, then by application of the functor Hom( ⊕ i∈I−I1 Pi,−) in the projective resolution 0 // Pm // · · · // P2 // P1 // P0 // S // 0 we will have, 0 // P̃m // · · · // P̃2 // P̃1 // P̃0 // S̃ // 0 so pdΓ1(S̃) ≤ m because rad2 A(P (Ω(S))) = 0 and P (Ω(S̃)) = P ( ˜Ω(S)) = P (Ω(S)) and ac- cording to Proposition 2.1, rad2 Γ1 (Hom( ⊕ i∈I−I1 Pi, P (Ω(S)))) ⊆ Hom( ⊕ i∈I−I1 Pi, rad 2 A(P (Ω(S)))) and the third assertion is verified. 5.2 Remark. The Simple A-modules (Si)i∈I−I1 have finite projective dimensions so the Γ1-simple modules are also simple, and by the third assertion in Proposition 3.1 and by Lemma 1.2, the Γ1-simple module S̃i2 of minimal projective dimension among the simple modules in modΓ1 checks pdΓ1(S̃i2) ≤ 1 and Ext1Γ1 (S̃i2 , S̃i2) = 0 by Theorem 1.3. 5.3 Theorem. If S̃i2 is the Γ1-simple module of minimal projective dimension among the simple modules of modΓ1, then Ext1A(Si2 , Si2) = 0 Proof. S̃i2 is a simple Γ1-module and S̃i2 = Hom( ⊕ i∈I−I1 Pi, Si2) with Si2 = Aei2/Jei2 where ei2 is a primitive idempotent a simple A -module not isomorphic to Si1, then Ext1A(S̃i2 , S̃i2) = 0 with pdΓ1(S̃i2) = 0 or 1. M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1861 Thus, • If S̃i2 is projective, then Si2 is projective and Ext1A(Si2 , Si2) = 0. • If S̃i2 is not projective, then Ext1A(Si2 , Si2) ̸= 0, therefore Hom(Jei2 , Si2) ̸= 0 and the exact sequence, 0 −→ Jei2 −→ Pi2 −→ Si2 −→ 0 will not be split, by the same pdA(Si2) ≥ 2 because i2 /∈ I1 and by application of the functor Hom( ⊕ i∈I−I1 Pi,−), we will have the exact sequence, 0 −→ Hom( ⊕ i∈I−I1 Pi, Jei2) −→ Hom( ⊕ i∈I−I1 Pi, Pi2) −→ Hom( ⊕ i∈I−I1 Pi, Si2) −→ 0 and since Hom( ⊕ i∈I−I1 Pi, Jei2) is not projective, then pdΓ1(Hom( ⊕ i∈I−I1 Pi, Si2) > 1 i.e. pdΓ1(S̃i2) > 1 which is absurd, thus Ext1A(Si2 , Si2) = 0 Now, to deduce the proof of our main contribution, namely Theorem 1.1. In fact, by induction and in the same way, we consider the Artin algebras, Γk = End( ⊕ i∈I−I1∪I2∪I3...∪Ik Pi) op where Ik ⊆ I such as top(Pi) ≃ Sik+1 for all index i ∈ I−I1∪I2∪I3...∪Ik and Sik+1 be a simple module of minimal projective dimension among the simple modules in modΓk for all i ∈ I − I1 ∪ I2 ∪ I3... ∪ Ik, and Ext1A(Sik , Sik) = 0, then we get the result. 6. Theoretical application as a corollary 6.1 Corollary. Let A be an Artinian ring with radical cubed zero such that the projective cover of rad(A) is of Loewy length two. If gdim(A) is finite, then Ext1A(S, S) = 0 for every simple module S. Proof. If rad2(P (rad(A)) = 0, then rad2(P (Ω(S))) = 0 for all simple A-modules indeed A/rad(A) = ⊕ i∈J Sai i where the (Si)i∈J are all the A non-isomorphic simple modules and as we have the exact sequence, 0 // rad(A) // A // A/rad(A) // 0 then, Ω(A/rad(A)) = rad(A) and if rad2(P (rad(A))) = 0, then, rad2(P (Ω(A/rad(A)))) = 0, then we have rad2(P (Ω(Si))) = 0 and according to the The- orem 3.4, we obtain Ext1A(S, S) = 0. M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1862 7. Discussions by practical examples 7.1. Couterexample to Theorem 2.3 The condition pd(S) ≤ 1 is not necessary but its consideration let us be sure to get Ext1A(S, S) = 0 which is the more practical result as it leads to the existence of the arrows in the extension quiver. This is just to say that we should not deny that there may be some particular cases where we may have Ext1A(S, S) = 0 even if pd(S) > 1 as we can illustrate in the following example. 7.1 Example. Consider the quiver 1 2 3 4 5a b c d and let A = KQ/I be its quiver algebra bounded by I =< abc, cd >. We have the list of all projective and indecomposable modules, P (1) : K −→ K −→ K −→ 0 −→ 0 P (2) : 0 −→ K −→ K −→ K −→ 0 P (3) : 0 −→ 0 −→ K −→ K −→ 0 P (4) : 0 −→ 0 −→ 0 −→ K −→ K P (5) : 0 −→ 0 −→ 0 −→ 0 −→ K and also, S(1) : K −→ 0 −→ 0 −→ 0 −→ 0 is the simple module corresponding to vertices 1 which is injective and coincides by I(1). Then, Ext1A(S(1), S(1)) = 0 because S(1) is an injective module and we have, 0 −→ P (5) −→ P (4) −→ P (2) −→ P (1) −→ S(1) −→ 0 which implies that pd(S) = 3 7.2. Weak no loop conjecture In what follows, we provide an example for the resolution of the weak no loop con- jecture, by taking a quiver algebra A verifying J3 = 0 and without considering that rad2(P (Ω(S))) = 0 for every simple module and that is to say that we may have some of these modules checking this condition. This is introduced just to help the reader to get used and understand the steps followed in our main final example and which the strong no loop conjecture. 7.2 Example. Consider the quiver defined by: M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1863 4 1 2 3 5 a b c e d and let A = KQ/I its quiver algebra bounded by I =< ab, ac > It’s clear that any path of length 3 is null therefore rad3(A) = 0 and we listed the way how to find the resolution projectives of any simple A-module in our example: we have 0 P (1) : K K 0 0 K2 P (2) : 0 K K2 K2 K P (3) : 0 0 K K K P (4) : 0 0 0 0 0 P (5) : 0 0 0 K M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1864 and we also have, 0 −→ Ω(S(1)) −→ P (1) −→ S(1) −→ 0 with, 0 Ω(S(1)) = S(2) : 0 K 0 0 and we can check that rad2(P (Ω(S(1))) ̸= 0. Then, we have the exact sequence, 0 −→ Ω(S(2)) −→ P (2) −→ Ω(S(1)) −→ 0 with, K2 Ω(S(2)) : 0 0 K2 K2 and K2 rad(Ω(S(2))) : 0 0 0 K2 then, Ω(S(2))/rad(Ω(S(2))) = S(3)2, and we have the exact sequence: 0 −→ Ω2(S(2)) −→ P (3)2 −→ (S(3))2 −→ 0 with K2 Ω2(S(2)) : 0 0 0 K2 M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1865 then we have, 0 −→ P (4)2 ⊕ P (5)2 −→ Ω2(S(2)) −→ 0 and finally the projective resolution of S(1) is, 0 −→ P (4)2 ⊕ P (5)2 −→ (P (3))2 −→ P (2) −→ P (1) −→ S(1) −→ 0 In the same way, we get, 0 −→ P (4)2 ⊕ P (5)2 −→ (P (3))2 −→ P (2) −→ S(2) −→ 0 0 −→ P (4) ⊕ P (5) −→ P (3) −→ S(3) −→ 0 0 −→ P (4) −→ S(4) −→ 0 0 −→ P (5) −→ S(5) −→ 0 By using the formula, Hom(P (i), S(j)) = Hom(Aei, S(j)) ≃ eiS(j) = { K if j = i 0 if j ̸= i and by application of the functor Hom(−, S(1)) to the resolution P∗(S(1)), i.e. the complex obtained by eliminating of S(1) in the resolution projective of S(1) we find the complex: 0 −→ K −→ 0 −→ 0 −→ ... and Ext1A(S(1), S(1)) = 0. Thus, by the same way, Ext1A(S(i), S(i)) = 0 for every 1 ≤ i ≤ 5. 7.3. Strong no loop conjecture without necessary specification of the nilpotence index of the Jacobson radical The following example clearly shows that if the extension quiver has a loop in a simple module then its projective dimension, is infinite although it is the nilpotence index of the Jacobson radical 7.3 Example. Let Q the quiver defined by 1 2 α β γ and let A = KQ/I which is algebra bounded by the relations α2 − βγ, γαβ and γβ. By definition, in the quiver algebra, we get Ext1A(S(1), S(1)) = 0. Then, by further calculations, we can find the infinite resolution of S(1) as following, ... −→ P (2) −→ P (1) −→ P (1) ⊕ P (2) −→ P (1) −→ S(1) −→ 0 M. Laaraj, S. Abdelalim, I.Elmouki / Eur. J. Pure Appl. Math, 17 (3) (2024), 1855-1868 1866 with, P (1) : K4 K2 and, P (2) : K2 K 7.4. Strong no loop conjecture finally resolved in J3 = 0 In this paper, we have succeed to resolve the strong no loop conjecture posed in [14] but here in the special case of J3 = 0 under the condition that the radical’s square of the cover projective of the first syzygy is zero, while using more tractable background focusing on non-commutative algebra and homology instead of some combination with K-theory as used in the past authors in [9] to solve this conjecture at least in the case of J2 = 0, however, the problem has remained open since that time for the case of J3 = 0 and which is of course very important to resolve for getting some inspiration and try to resolve further cases like J4 = 0 or more generally Jn = 0, either by introducing one con- dition like we have done here by taking a zero radical’s square of the cover projective of the first syzygy, or maybe some would think about more additional conditions in the future. In what follows we will present an example of an Artin algebra with Jacobson cube radical 0 and that also satisfies the condition of the First lemma. 7.4 Example. Consider the algebra A = KQ/I given over a field K by a quiver Q and an admissible ideal I =< bc, ad > as following: 3 1 2 5 4 c α β a b d Clearly, any path in A of length 3 is zero, and therefore rad3(KQ/I) = 0. Here are the calculations of the projective resolutions of simple modules (S(i))1≤i≤5, 0 −→ P (3)⊕ P (4) −→ P (2)2 −→ P (1) −→ S(1) −→ 0 REFERENCES 1867 0 −→ P (3)⊕ P (4) −→ P (2) −→ S(2) −→ 0 0 −→ P (5) −→ P (3) −→ S(3) −→ 0 0 −→ P (5) −→ P (4) −→ S(4) −→ 0 0 −→ P (5) −→ P (4) −→ S(5) −→ 0 Then, we have inf(pd(Si))1≤i≤5 = pd(S2) = pd(S3) = pd(S4) and rad2(P (Ω(S))) = 0, where P (Ω(S)) the projective cover for Ω(S) for every simple module S. Thus, Ext1A(S(i), S(i)) = 0 for i = 2; i = 3 and i = 4. 8. Conclusion Our work ultimately resolved the conjecture for Artinian rings by relying on our main results and that could be summarized on the statement of Theorem 2.1 and which we have succeeded to prove along this paper, as well as the statement of Corollary 6.1. which gen- eralizes it, and finally without forgetting our other contribution point stated in Remark 1.1. We may follow similar process in the future in the hope to demonstrate the conjecture in the case of zero radical cube without a condition on the syzygies but also extending this research by attacking the conjecture in the case of J4 = 0 under constraints on the syzygies. The whole approach may lead to a process of proving the conjecture in general. Acknowledgements A special thanks to the editor Professors Eyup Cetin and Baris Kiremitci. We would also like to thank all the three anonymous referees for their time, effort and help for improving the content of our paper. References [1] I. Elmouki S. Abdelalim. 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