EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 1197-1205 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the self-injectivity and CM-free of an Artin algebra with radical cubic zero Mounir Laaraj1,∗, Seddik Abdelalim1 1 Laboratory of Fundamental and Applied Mathematics (LMFA), Faculty of Sciences Ain Chock (FSAC), University Hassan II of Casablanca (UNIVH2C), Morocco. Abstract. In this paper, we succeed in proving that a connected Artin algebra whose Jacobson cubic radical is zero with each simple module and each Gorenstein projective module having the square radical of the cover projective of its first syzygy to be zero and verifying the coincidence covers, is either self-injective or CM-free. 2020 Mathematics Subject Classifications: 16P20, 16P10, 16G10, 16N80, 13H10, 16E30 Key Words and Phrases: Artin algebras, Representation-finite algebras, Homological algebra, Gorenstein algebra, Jacobson radical Introduction Let K be a commutative ring and A an Artin algebra over K of Jacobson radical with nilpotency index 3, mod(A) will be the category of finitely generated modules over A and all modules in this work are in left in the category mod(A), we know that the non-isomorphic simple A-modules are in finite number and denoted (Si)i∈I allowing us to define the extension quiver denoted QA according to the condition Ext1A(Si, Sj) ̸= 0 by defining an arrow from Si to Sj thus A is connected if and only if the algebra QA is connected. We also recall that ExtnA(M,N) = Hn(HomA(P∗, N)) is the nth cohomology of the cochain complex of K-modules HomA(P∗, N) which is : · · · // 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 , where A is a K-algebra with K is a commutative ring. In 2012, Xiao-Wu Chen classified the algebras whose radical square is zero and showed ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5085 Email addresses: mounirlaaraj2@gmail.com (M. Laaraj), seddikabd@hotmail.com (S. Abdelalim) https://www.ejpam.com 1197 © 2024 EJPAM All rights reserved. M. Laaraj, S. Abdelalim / Eur. J. Pure Appl. Math, 17 (2) (2024), 1197-1205 1198 that they are either self-injective or CM-free, see [5]. However, this result has never been proven in the case of cube radical 0 and the same author pointed out that it is false using a counter-example with a non-zero square radical algebra. One year later, Luo Rong proved that all local algebras with radical cubic zero are PCM- free, see [2] . An important invariant in our subject is the radical square of the projective cover of the first syzygy of each simple A-module and each Gorenstein-projective A-module that we consider here to be 0, so by a formula characterizing the calculation of the radical of syzygy of an indecomposable A-module of Lowey length 2, that is to say from the square radical 0, we will extend this calculation to a Gorenstein-projective A-module and by char- acterization of the latter, we construct a family (Si)i∈J of simple A-modules non-projective and Gorenstein-projective and the family (Pi)i∈J of projective modules such that each Pi is associated to Si. The quiver QA will be connected, which implies the self-injectivity for the algebra A with zero radical cubic. Since that assumption is not always true, we succeed by using our condition, to establish that the property of auto-injectivity and CM-free remain valid. If M is an A-module, we note Ω(M) as the first syzygy of M and P (M) as the projective cover of M with Ω(M) is the kernel of the essential epimorphism: P (M) −→ M ie P (M)/rad(M) ≃ M/rad(M) see [4], similarly we note Ω2(M) = Ω(Ω(M)). In the upcoming parts of the paper, we start by discussing some properties of the Gorenstein- projective modules, then in the case of an Artin algebra A whose Jacobson cubic radical is zero while considering a non-projective and indecomposable Gorenstein-projective, we characterize the radical of syzygy of an A-module. Finally, we prove by using our condi- tion, that the properties of self-injectivity and CM-free remain true. 1. Gorenstein-projective modules 1.1 Definition. Recall that the infinite sequence of A- modules (Cn)n∈Z : · · · // Cn−1 dn−1 // Cn dn // Cn+1 dn+1 // · · · is called complex and denoted C⋆ if Im(dn−1) ⊆ Ker(dn). Denote Bn(C⋆) = Im(dn−1), Zn(C⋆) = Ker(dn) and Hn(C⋆) = Zn(C⋆)/B n(C⋆) where Zn(C⋆) is called cocycle and the complex C⋆ is called acyclic if it is an exact sequence, i.e. Hn(C⋆) = 0 for all n. 1.2 Remark. If the Cn’s are projective for all n and Hom(C⋆, A) is acylic, then C⋆ is called totally acyclic. M. Laaraj, S. Abdelalim / Eur. J. Pure Appl. Math, 17 (2) (2024), 1197-1205 1199 1.3 Definition. An A-module M in mod(A) is said Gorenstein-projective provided that there is a totally acyclic complex (P⋆) of projective modules such that its 0-th cocycle Z0(P⋆) is isomorphic to M . 1.4 Example. - Any A-module P projective is a Gorenstein-projective module; just con- sider the exact sequence P⋆: · · · // 0 // P IdP // P // 0 // · · · and we have Z0(P⋆) ≃ P . - Over the algebra Z/4Z, the complex: · · · // Z/4Z // Z/4Z // Z/4Z // · · · defined by multiplication by 2 is acyclic and it remains exact by applying the functor Hom(−, P ) with P a projective module, then 2Z/4Z is a Gorenstein-projective but not projective as Z/4Z-module. 1.5 Remark. The full subcategory of the finite type A-module category constituted by the Gorenstein-projective modules is denoted A-Gproj and the subcategory of projective A-modules is included in A-Gproj. Denote by ⊥A the full subcategory ofA-mod consisting by modulesM such that ExtiA(M,A) = 0 for all i ≥ 1. Recall the following lemma, see [3]. 1.6 Lemma. Let (P⋆) be a complex of projective A-modules. Then the following statements are equivalent: (1) the complex (P⋆) is totally acyclic. (2) the complex (P⋆) is acyclic and each cocycle Zi(P⋆) lies in ⊥A ; (3) the complex Hom((P⋆), A) is totally acyclic. We note we will need the corollary and proposition stated in [1] thereafter which says the following, 1.7 Corollary. If M is a Gorenstein-projective module, then Ext1A(M,L) = 0 for all module L of finite projective dimension. 1.8 Proposition. Let M in mod(A), then M is a Gorenstein-projective module if and only if there exists a long exact sequence: 0 //M // P0 // P1 // P2 // · · · with Pi are the projective modules and each cocycle in ⊥A. Proof. By definition M is in A-Gproj then, there exists a totally acyclic complex P⋆ such that M ≃ Z0(P⋆) and we have: 0 //M // P0 // P1 // P2 // · · · M. Laaraj, S. Abdelalim / Eur. J. Pure Appl. Math, 17 (2) (2024), 1197-1205 1200 . Now for the reciprocal, we consider · · · // Q2 // Q1 // Q0 //M // 0 the projective resolution ofM and by spiling the two resolutions, we get an acyclic complex P such that M ≃ Z0(P⋆). 1.9 Proposition. Let ε : 0 // X // Y // Z // 0 be a short exact sequence of A-modules. Then we have the following statements: (1) if X, Z are Gorenstein-projective, then so is Y . (2) if Y , Z are Gorenstein-projective, then so is X. Proof. Since X and Z are Gorenstein-projective modules, then by Proposition 1.8, we have two monomorphisms: 0 // X iX // X ′ and 0 // Z iZ // Z ′ such that X ′ and Z ′ are projective modules and the cokernels X1 and Z1 of iX and iZ respectively are Gorenstein-projective. Since by Corollary 1.7, Ext1A(Z,X ′) = 0 and by applying the functor Hom(−, X ′) to ε , we infer that the induced map: HomA(Y,X ′) HomA(f,X′)// HomA(X,X ′) is epimorphism, then there exists a morphism a : Y −→ X ′ such that: a ◦ f = IdX therefore we have the following exact diagram: 0 // X f // iX �� Y g // a iZ ◦ g  �� Z // iZ �� 0 0 // X ′ 1 0  // X ′⊕Z ′( 0 1 ) // Z ′ // 0 By the Snake lemma, we have a short exact sequence: 0 // coker(iX) // Y1 // coker(iZ) // 0 Therefore 0 // X1 // Y1 // Z1 // 0 where Y1 = ( a iZ ◦ g ) and since X1 and Z1 are in ⊥A we infer that Y1 in ⊥A and by iterating this process and using Proposition 1.8, we show that Y is Gorenstein-projective. For a second statement, we take the exact sequence: 0 // Z ′ // P // Z // 0 M. Laaraj, S. Abdelalim / Eur. J. Pure Appl. Math, 17 (2) (2024), 1197-1205 1201 such that P is projective and Z ′ is Gorenstein-projective, and by the following pullback diagram: 0 �� 0 �� Z ′ �� Z ′ �� 0 // X // �� U // �� P // �� 0 0 // X // Y // �� Z // �� 0 0 0 We have U a Gorenstein projective module by using (1) in a middle column and by the first row,U ≃ X ⊕ P since P is projective we infer that X is Gorenstein projective. Recall for a left A-module AM over an Artin algebra, there exists a projective module denoted P (M) such that the kernel of the epimorphism f : P (M) −→ M is superfluous in P (M) that to say ker(f) ⊆ rad(P (M)). Also, this epimorphism is called the projective cover of M and we have a short exact sequence: 0 // Ω(M) // P (M) //M // 0 1.10 Corollary. Let M be an A-module. If M is Gorenstein-projective, then so are Ω i(M) for i ≥ 1. Proof. Take a short exact sequence : 0 // Ω(M) // P (M) //M // 0 by applying Proposition 1.9 and since P (M) is projective, then Ω(M) are likewise, and by iterating this process in the short exact sequence: 0 // Ω i+1(M) // P (Ω i(M)) // Ω i(M) // 0 we have the result. For the following work, we need the following lemma, see [1, 2, 6]. 1.11 Lemma. Let M be a non-projective and indecomposable Gorenstein-projective A- module, the A-module Ω(M) is also non-projective indecomposable, and Gorenstein-projective. 2. Radical of syzygy of an A-module In what follows, A is an Artin algebra whose Jacobson cubic radical is zero. M. Laaraj, S. Abdelalim / Eur. J. Pure Appl. Math, 17 (2) (2024), 1197-1205 1202 2.1 Lemma. Let M be a non-projective module in modA with rad2(M) = 0 and ΩM indecompasable. If f : P (M) → M is a projective cover of M , then: rad(ΩM) = rad2(P (M)). 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. Since now ΩM is indecomposable, there is a contradiction and the proof of the lemma is complete. 2.2 Corollary. Let M be a non-projective and indecomposable Gorenstein-projective A-module in modA with rad2(M) = 0, then: rad(ΩM) = rad2(P (M)) 3. The self-injectivity and CM-free algebras Recall that an Artin algebra A is said to be CM-finite if, up to isomorphism, there are only a finite number of indecomposable modules in A-Gproj, and this algebra is said to be CM-free if each indecomposable Gorenstein-projective module is projective. The following theorem is the extension of Theorem 2.3.9 quoted by [2] in the case rad3(A) = 0 under some condition and the generalization is false as the author has given this coun- terexample: LetA = [ K[X]/(X2) K[X]/(X2) 0 K[X]/(X2) ] be the Artin algebra, it is easy to show that rad3(A) = 0 and rad2(A) ̸= 0. Also, it is not self-injective . Our work shows that the result is true in a particular case for a specific algebra that we will introduce its definition, 3.1 Definition. Let A be an artin algebra, we say that A verifies the coincidence covers if, P (Ω(S)) ≃ P (Ω(T )) implies S ≃ T for all S and T simples modules. 3.2 Theorem. Let A be a connected Artin algebra with radical cubed zero, and rad2(P (ΩM)) = 0 for any A-module M either simple or Gorenstein-projective and verifying the coincidence covers, then A is self-injective or CM-free. Proof. Suppose that A is not CM-free, then there exists a non-projective indecompos- able A-module M and Gorenstein-projective, and we have a short exact not-split sequence: 0 //M f // P (M) p //M ′ // 0 M. Laaraj, S. Abdelalim / Eur. J. Pure Appl. Math, 17 (2) (2024), 1197-1205 1203 and we have M ′ = coker(f) is a Gorenstein-projective A-module and p is the projective cover of M ′, then Ω(M ′) = M and since Ω(M ′) ⊆ rad(P (M)), then rad2(Ω(M ′)) = 0, i.e. rad2(M) = 0 and since M is non-projective and indecomposable, then by Corollary 2, we have rad(Ω(M)) = rad2(P (M)) = rad2(P (Ω(M ′))) = 0 , then Ω(M) is semisimple and since Ω(M) is indecomposable, the A-module Ω(M) is simple. Let S1 = Ω(M) and take the short not-split exact sequence: 0 // Ω(S1) i2 // P0 p // S1 // 0 We consider S2 = Ω(S1), then Ext1A(S1, S2) ̸= 0 and we are finding an arrow in S1 to S2 because S2 is non-projective and indecomposable, i.e. S2 is a simple module. In fact, we show that the only arrow having the target S2 has S1 for vertices in the contrast case, assume that there exists a simple module S such that Ext1A(S, S2) ̸= 0 and consider the projective resolution of S is as follows: 0 // Ω2(S) // P1(S) // P0(S) // S // 0 We know that, 0 // Ω(S) i0 // P0(S) // S // 0 and 0 // Ω2(S) // P1(S) // Ω(S) // 0 are the short exact sequences characterezing the first and second syzygy of S, then Ω2(S) ⊆ rad(P1(S)) and since rad2(P1(S)) = 0, then rad(Ω2(S)) = 0, then Ω2(S) is a semisimple module. Since Ext1A(S, S2) ̸= 0 and Hom(Ω(S), S2) ̸= 0 so, there exists an epimorphism f ̸= 0 in Hom(Ω(S),Ω(S1)) and we get the diagram: 0 // Ω2(S) i // g �� P π // h �� Ω(S) // f �� 0 0 // Ω2(S1) i1 // P1 π1 // k << Ω(S1) // 0 , then f ̸= 0 implies h ̸= 0 if not we get f ◦ π = π1 ◦ h = 0 absurd, so because f ̸= 0 and Ω(S1) is a simple module, then f is an epimorphism, and since P1 is a projective module, then there exists a non-zero morphism K : P1 −→ Ω(S) such that f ◦ k = π1, if one could prove that i0 ◦ k is an epimorphism, that would be a great proof, however, this is not that easy since the only information that we have, is that i0 is injective. This point will be true through the Gorenstein projective, but this is very difficult to prove even if the existence of such epimorphism looks right by using this property, as in the end i0 ◦ k being an epimorphism will imply a contradiction that S = 0 Thus, we choose to proceed as follows. REFERENCES 1204 As Ω2(S) is semisimple and Ω2(S1) is simple and g ̸= 0, then Ω2(S1) is a direct summand of Ω2(S), so we can inject Ω2(S1) into Ω2(S) which in turn is injected into P , let l : Ω2(S1) ↪→ P be the injection composed of these last injections and with the previous notations and in short we have l ∈ Hom(S3, P ) with S3 = Ω2(S1) and by application of the functor Hom(−, P ) to the short exact sequence, 0 // S3 i1 // P1 π1 // S2 // 0 we will have, 0 // Hom(S2, P ) // Hom(P1, P ) Hom(i1,P )// Hom(S3, P ) // Ext1A(S2, P ) = 0 and this because S2 is a simple and Gorenstein projective A-module, so l = Hom(i1, P )(a) = a ◦ i1 with a ∈ Hom(P1, P ) and note that S3 is the socle of P1 on which a is non-zero so a is a monomorphism and length(P1) ≤ length(P ) and as f and g are epimorphisms, then h is also and length(P ) ≤ length(P1), therefore, P ≃ P1 that implies that Ω(S1) ≃ Ω(S), and like A verifying the coincidence covers see Definition 3.1, therefore S1 ≃ S. Finally, our construction affirms the existence of the set {S1, S2, ..., Sn−1} of pairwise non-isomorphic simple A-modules; moreover, each Si satisfies that any simple A-module S with Ext1A(S, Si) ̸= 0 is isomorphic to Si−1 and we considered Si+1 = Ω(Si), then we have any simple A-module S with Ext1A(Si, S) ̸= 0 is isomorphic to Si+1 and by identification S0 with Sn−1 and Sn with S1 it follows that the full subquiver QA with vertices {S1, S2, ..., Sn−1} is a connected . Since the algebra A is connected and since there are simple A-modules associated with all the indecomposable projective A-modules given by {P1, P2, ..., Pn−1} it follows by Theorem 9.3.7 [4] that the algebra A is self-injective. Acknowledgements A special thanks to the editor Professor Dr. Eyup Cetin and Dr. Ilias Elmouki. 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] Xiao-Wu Chen. 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