Local Systems of Simple Locally Finite Associative Algebras
aa r X i v : . [ m a t h . R A ] J a n LOCAL SYSTEMS OF SIMPLE LOCALLY FINITE ASSOCIATIVEALGEBRAS
HASAN M. S. SHLAKA
Abstract.
In this paper, we study local systems of locally finite associative algebrasover fields of characteristic p ≥ . We describe the perfect local systems and studythe relation between them and their corresponding locally finite associative algebras. -perfect and conical local systems are also be considered and described briefly. Introduction
Throughout the paper the field F is algebraically closed of characteristic p ≥ and A is an infinite (countable) dimensional locally finite associative algebra over F . Recallthat an algebra A is called locally finite if every finite set of elements is contained in afinite dimensional subalgebra of A [1]. A is called locally simple if for any finite subset U of A , there is a finite dimensional simple subalgebra that contains U . Note that wedo not require A to have an identity element.Locally finite Lie algebras were described by Bahturin and Strade [3] in 1995. Theydescribed these algebras in terms of local systems of locally finite Lie algebras. Recallthat a system of finite dimensional subalgebras { A α } α ∈ Γ of an algebra A over F is saidto be a local system for A if A = ∪ α ∈ Γ A α and for each α, β ∈ Γ , there is γ ∈ Γ suchthat A α , A β ⊆ A γ . Bahturin and Strade in [2] provided some examples to describelocally finite Lie algebras. In several papers (see for example [4], [5], [7] and [11])Baranov et. al. classified simple locally finite Lie algebras over algebraically closedfields of characteristic zeros. They showed that there are two classes of locally finiteLie algebras which have can be characterized in many different ways. These are thesimple diagonal locally finite Lie algebras and the finitary simple Lie algebras. Innerideals of the finitary simple Lie algebras were studied by Fernandiz Lopes, Garcia andGomez Lozano in [12], while inner ideals of the other class were studied by Baranovand Rowley in [9].In 2004, Bahturin, Baranov and Zalesski [1] studied the simple locally finite associ-ative algebras over algebraically closed fields of zero characteristic. They highlightedthe relation between them and locally finite Lie algebras over algebraically closed fieldof characteristic 0. They proved that simple Lie subalgebras of locally finite associativeones are either finite dimensional or isomorphic to the Lie algebra of skew symmetricelements of some (Type1) involution simple locally finite associative algebras.Baranov in [6] proved that there is a natural bijective correspondence between suchLie algebras and locally involution simple associative algebras over algebraically closedfields of any characteristic = 2 . Thus, to classify locally finite Lie algebras, we need to study locally finite associative algebras briefly. This requires a good understanding oftheir local systems.In this paper, we study local systems of locally finite associative algebras over fieldsof characteristic p ≥ . Some of the results in this papers were mentioned in [1] in thecase when p = 0 . We start with some of the preliminaries in section two. Section threeis devoted to the study of local systems of locally finite associative algebras.In section four, we describe the perfect local systems and study the relation betweenthem and their corresponding locally finite associative algebras. Finally, the -perfectand conical local systems were described.2. Preliminaries
Recall that F is an algebraically closed field of characteristic p ≥ . Definition 2.1. [1] A locally finite algebra is an algebra A over a field F in which everyfinite set of elements of A is contained in a finite dimensional subalgebra of A .As an example of locally finite associative algebra is the algebra M ∞ ( F ) of infinitematrices with finite numbers of non-zero entries, that is,(2.1) M ∞ ( F ) = ∞ ∪ n =1 M n ( F ) , where the algebra M n ( F ) can be embedded in M ( n +1) ( F ) by putting M n ( F ) in the leftupper hand corner and bordering the last column and row by ’s. Definition 2.2.
Let A be an algebra over a field F . A system of finite dimensionalsubalgebras { A α } α ∈ Γ of A is said to be a local system for A if A = ∪ α ∈ Γ A α and foreach α, β ∈ Γ , there is γ ∈ Γ such that A α , A β ⊆ A γ .Put α ≤ β if A α ⊆ A β . Then Γ is a directed set and A = lim −→ A α is a direct limit ofthe algebras A α . Recall that A is called perfect algebra if AA = A. Definition 2.3.
1. A local system is said to be perfect (resp . simple, semisimple,nilpotent, ... etc ) if it consists of perfect (simple, semisimple, nilpotent, ... etc) algebras.2. A locally finite algebra A is called locally perfect (resp . simple, semisimple,nilpotent, ... etc ) if it consists a perfect (simple, semisimple, nilpotent, ... etc) localsystem. Example 2.4.
Suppose that A is locally simple. Then there is a chain of simplesubalgebras A ⊂ A ⊂ A ⊂ . . . of A such that A = ∪ ∞ i =1 A i . We can view A as the direct limit lim −→ A i for the sequence A → A → A → . . . of injective homomorphisms of finite dimensional simple associative algebras A i . Since F is algebraically closed, each A i can be identified with the algebra M n i ( F ) of all n i × n i -matrices over F for some n i . Moreover, each embedding A i → A i +1 can be written inthe following matrix form M → diag ( M, . . . , M, , . . . , , M ∈ M n i ( F ) . IMPLE LOCALLY FINITE ASSOCIATIVE ALGEBRAS 3 Local Systems of Locally Finite Associative Algebras
Lemma 3.1.
Let A be a locally finite associative algebra over F and let { A α } α ∈ Γ be asystem of finitely generated subalgebras of A . Then { A α } α ∈ Γ is a local system of A ifand only if for every finite dimensional subspace P of A there exists β ∈ Γ such that P ⊆ A β .Proof. Suppose first that { A α } α ∈ Γ is a local system of A . Let P be a finitely generatedsubalgebra of A and let { p , . . . , p n } be a basis of P . Then A = ∪ α ∈ Γ A α and for each ≤ i ≤ n , there is A i ∈ { A α } α ∈ Γ such that p i ∈ A i . Thus P ⊆ A β for some β ∈ Γ , asrequired.Suppose now that for every finite dimensional subspace P of A there exists β ∈ Γ such that P ⊆ A β . We need to show that { A α } α ∈ Γ is a local system of A . Let x ∈ A .Then for every subspace P x generated by x , there is β ∈ Γ such that P x ⊆ A β , so A = ∪ α ∈ Γ A α . It remains to note that for any α, β ∈ Γ , there is γ ∈ Γ such that A α , A β ⊆ A γ . (cid:3) Lemma 3.2.
Suppose that { A α } α ∈ Γ is a local system of a locally finite associativealgebra A over F . If Γ = ∪ ri =1 Γ i , then { A α } α ∈ Γ k (for some ≤ k ≤ r ) is a local systemof A .Proof. Suppose that
Γ = ∪ ri =1 Γ i . We may assume that Γ is a is a disjoint union of Γ i of Γ (because if it is not, then we can decompose it as a disjoint union of subsets). We aregoing to prove by contradiction that { A α } α ∈ Γ k , for some ≤ k ≤ r , is a local system of A . Assume to the contrary that { A α } α ∈ Γ i is not a local system of A for each ≤ i ≤ r .Then there is a finite dimensional subspace P i of A such that P i / ∈ { A α } α ∈ Γ i for each i .Consider the subspace P = L ri =1 P i of A . Then P is finite dimensional with P * A α for all α ∈ Γ (because Γ is a disjoint union of the subsets Γ i ). Thus, { A α } α ∈ Γ is not alocal system of A , a contradiction. (cid:3) Lemma 3.3.
Let { A α } α ∈ Γ be a local system of A and let P be a finite dimensionalsubspace of A . Then { A β } β ∈ Γ P is a local system of A , where Γ P = { β ∈ Γ | P ⊆ A β } .Proof. By Lemma 3.1, there is β ∈ Γ such that A β ⊃ P . Let P be a finite dimensionalsubalgebra of A . Then P = P + P is a finite dimensional subalgebra of A , so byLemma 3.1, There is γ ∈ Γ such that A γ ⊃ P with γ ≥ β . Continuing with thisprocess we get the set Γ P = { β ∈ Γ | A β ⊃ P } ⊂ Γ . By Lemma 3.2, { A β } β ∈ Γ P is alocal system of A . (cid:3) Proposition 3.4.
Let { A α } α ∈ Γ be a local system of a simple locally finite associativealgebra A over F . The following hold.1. [13] Let { I α } α ∈ Γ be a system of ideals such that I α is an ideal of A α for each α ∈ Γ .Then either ∩ α ∈ Γ I α = 0 or for every k ∈ Γ or there exists β k ∈ Γ such that A k ⊆ I β k .2. Suppose that Γ P = { β ∈ Γ | P ⊆ A β } . Then { A Pβ } β ∈ Γ P is a local system of A ,where A Pβ is the ideal of A β that generated by the algebra P for all β ∈ Γ P .Proof.
1. This is proved in [13]. For the proof see [13, Proposition 4.5].2. By Lemma 3.3, { A β } β ∈ Γ P is a local system of A . Let A Pβ be the ideal of A β that generated by P . We need to show that { A Pβ } β ∈ Γ P is a local system of A . Since HASAN M. S. SHLAKA P ⊆ ∩ β ∈ Γ P A Pβ , we have ∩ β ∈ Γ P A Pβ = 0 , so by (1), for each β ∈ Γ P , there is γ ∈ Γ P suchthat A β ⊆ A Pγ . Thus, L = ∪ β ∈ Γ P A Pβ . It remain to note that A Pβ , A Pγ ⊆ A Pζ . Indeed,for each β, γ ∈ Γ P , there is ζ ∈ Γ P such that A β , A γ ⊆ A ζ because { A β } β ∈ Γ P is a localsystem of A . Therefore, { A Pβ } β ∈ Γ P is a local system of A , as required. (cid:3) Proposition 3.5.
Let A be a simple locally finite associative algebra and let { A α } α ∈ Γ be a local system of A . Let S be a non-zero finite dimensional subspace of A and let A sα be the ideal of A α generated by S . Then { A sβ } β ∈ Γ s is a local system of A , where Γ s = { β ∈ Γ | S ⊆ A β } .Proof. This follows directly from Proposition 3.4. (cid:3)
Recall that an associative algebra A is nilpotent if there is a positive integer n suchthat A n = 0 . Put A = A and A i = AA i − for all i > . Definition 3.6.
1. We say that an associative algebra P over a field F is residuallynilpotent if ∩ ∞ i =1 P i = 0 .2. We say that a locally finite associative algebra A is residually nilpotent if everyfinitely generated subalgebra P of A is residually nilpotent associative algebra. Theorem 3.7.
Let A be a simple locally finite associative algebra over F . Then everylocal system { A α } α ∈ Γ contains an algebra which is not residually nilpotent.Proof. Let { A α } α ∈ Γ be a local system of A . Consider the algebra P = F p , where p ∈ A is non-zero. Put Γ p = { α ∈ Γ | A α ⊃ P } . By Lemma 3.3, { A α } α ∈ Γ p is a local systemof A . Assume to the contrary that all A α are residually nilpotent for all α ∈ Γ . Thenfor each α ∈ Γ p , there is a positive integer n such that p ∈ A nβ and p / ∈ A n +1 β , so weget system of ideals { A nα } α ∈ Γ p with ∩ α ∈ Γ P A nα = 0 . By Proposition 3.4, for each α ∈ Γ p ,there is β ∈ Γ p such that A α ⊆ A nβ . Thus, A α ⊆ A n +1 β , so p / ∈ A α for all α ∈ Γ . Since p ∈ A (because A = A as A is simple), there exists γ ∈ Γ such that p = P ni =1 x i y i for some x i , y i ∈ A γ ( ≤ i ≤ n ). Thus, p ∈ A γ , a contradiction with p / ∈ A α foreach α ∈ Γ . Therefore, every local system of A must contain an algebra which is notresidually nilpotent, as required. (cid:3) Corollary 3.8.
No simple locally finite associative algebra can be locally residuallynilpotent.Proof.
This follows from Theorem 3.7. (cid:3) Perfect local Systems
Recall that a local system is said to be perfect (resp . simple, semisimple, nilpotent,... etc ) if it consists of perfect (simple, semisimple, nilpotent, ... etc) algebras.
Theorem 4.1.
Any simple locally finite associative algebra posses a perfect local system.Proof.
Let { A α } α ∈ Γ be a local system of A . Put A ∞ α = ∩ ∞ i =1 A iα . Then A ∞ α is a perfect subalgebra of A ∞ . Moreover, { A ∞ α } α ∈ Γ is a local system of thesubalgebra A ∞ = ∪ α ∈ Γ A ∞ α of A . Indeed, if α, β ∈ Γ , then there is γ ∈ Γ such that IMPLE LOCALLY FINITE ASSOCIATIVE ALGEBRAS 5 A α , A β ⊆ A γ , so A ∞ α , A ∞ β ⊆ A ∞ γ . Note that A ∞ is an ideal of A and A ∞ = 0 (byTheorem 3.7), but A is simple, so A ∞ = A . Therefore, A contains a perfect localsystem, as required. (cid:3) Lemma 4.2.
Let A be a simple locally finite associative algebra over F . The followingholds: (1) [3] Let { A α } α ∈ Γ be a local system of A . Then for every α ∈ Γ , there is ζ ∈ Γ such that A α ⊂ A ζ and A α T rad A ζ = 0 . (2) [9] If { A α } α ∈ Γ is a perfect local system of A , then there exists α ′ ∈ Γ for each α ∈ Γ such that rad A β T A α = 0 for all β ≥ α ′ .Proof.
1. Let A α ∈ { A α } α ∈ Γ . Suppose that Γ α = { β ∈ Γ | A α ⊆ A β } . By Lemma3.3, { A β } β ∈ Γ P is a local system of A . Let { rad A β } β ∈ Γ α be the system of the radicals of { A β } β ∈ Γ α such that rad A β is the radical of A β . By Theorem 3.7, not all members of { A β } β ∈ Γ P are nilpotent algebras, so there exists γ ∈ Γ α such that A γ * rad A β for all β ∈ Γ α . Thus, ∩ β ∈ Γ α rad A β = 0 (by Proposition 3.4).Now, we have A α ∩ rad A β ⊆ rad A α for all β ∈ Γ α . Since dim(rad A β ) is finite, thereexist β , β , . . . , β n ( n ≥ ) such that A α ∩ ( ∩ ni =1 rad A β i ) = 0 . Now, take any ζ ∈ Γ α such that A ζ ⊇ A β , . . . , A β n . Then A β i ∩ rad A ζ ⊆ rad A β i for all ≤ i ≤ n , so A α ∩ rad A ζ ⊆ A β i ∩ rad A ζ ⊆ rad A β i , for all ≤ i ≤ n. Therefore, A α ∩ rad A ζ ⊆ A α ∩ ( ∩ ni =1 rad A β i ) = 0 , as required.2. This follows from part (1.). (cid:3) Theorem 4.3.
Let { A α } α ∈ Γ be a local system of a simple locally finite associativealgebra A over F . Then for each α ∈ Γ , there is γ ∈ Γ such that A α ⊂ A γ and M γ ∩ A α = 0 for some maximal ideal M γ of A γ .Proof. By Lemma 4.2, A has a perfect local system. Suppose that { A α } α ∈ Γ is a perfectlocal system of A . Fix any α ∈ Γ . By Lemma 4.2(2), there is ζ ∈ Γ such that(4.1) A α ⊂ A ζ and A α ∩ rad A ζ = 0 . Let Γ ζ = { β ∈ Γ | β ≥ ζ } . Then by Lemma 3.2, { A β } β ∈ Γ ζ = { A β ∈ { A α } α ∈ Γ | A ζ ⊂ A β } is a local system of A . Let { P ζ , . . . , P ζ t } and { M ζ , . . . , M ζ r } be the set of all minimalperfect and maximal ideals of A ζ , respectively. Consider two subsets Γ ζl,k and Γ ζ of Γ ζ defined as follows: Γ ζk,l = { β ∈ Γ ζ | A β has an ideal I β with P ζ k ⊂ I β ∩ A ζ ⊂ M ζ l , ≤ k ≤ t, ≤ l ≤ r } ;Γ ζ = { β ∈ Γ ζ | any proper ideal I β of A β satisfies I β ∩ A ζ ⊂ rad A ζ . Then Γ ζ = Γ ζk,l ∪ Γ ζ , so by Lemma 3.2, either { A β } β ∈ Γ ζk,ℓ or { A β } β ∈ Γ ζ is a local systemof A . We claim that { A β } β ∈ Γ ζ is a local system of A . Assume to the contrary that { A β } β ∈ Γ ζk,l is local system. Then for any A γ ∈ { A β } β ∈ Γ ζk,l , there is an ideal I γ ⊂ A γ such HASAN M. S. SHLAKA that P ζ k ⊂ I γ ∩ A ζ ⊆ M ζ l for some k and l . Hence, we get a system of ideals { I β } β ∈ Γ ζk,ℓ such that I β is an ideal of A β with P ζ k ⊂ I β ∩ A ζ ⊂ M ζ l for some ≤ k ≤ t and ≤ l ≤ r .Note that I_ β + A ζ for all β ∈ Γ ζk,l (because if A ζ ⊆ I β , then A ζ = I β ∩ A ζ ⊂ M ζ l , acontradiction as M ζ l is a proper maximal ideal of A ζ . Since A ζ ⊆ A β , for each β ∈ Γ ζk,l ,there is no δ β ∈ Γ ζk,l such that A β ⊆ I δ β , so by Proposition 3.4, ∩ β ∈ Γ ζk,l I β = 0 , but ∩ β ∈ Γ ζk,l I β ∈ { I β } β ∈ Γ ζk,l , so there is some ≤ k ≤ t such that = P ζ k ⊂ ∩ β ∈ Γ k,l I β , a contradiction.Hence, { A β } β ∈ Γ ζ is a local system of A . Thus, for every proper ideal I γ of A γ ∈{ A β } β ∈ Γ ζ , we have I γ ∩ A ζ ⊆ rad A ζ , so for α ∈ Γ , there is γ ∈ Γ ζ ⊂ Γ ζ ⊂ Γ such thatif I γ is a proper ideal of A γ , then I γ ∩ A α = ( M γ ∩ A ζ ) ∩ A α ⊂ rad A ζ ∩ A α = 0 . Therefore, for each α ∈ Γ , there exists γ ∈ Γ such that A α ⊂ A γ and M γ ∩ A α = 0 forsome maximal ideal M γ of A γ , as required. (cid:3) If A is finite dimensional, then it follows by Wedderburn-Malcev Theorem (see [8,Theorem 1]) that there exists a semisimple subalgebra S of A such that A = S ⊕ rad A and for any semisimple subalgebra Q of A , there is r ∈ rad A with Q ⊆ (1 + r ) S (1 + r ) . Theorem 4.4. [8, Theorem 6]
Let A be a finite dimensional algebra and let I be a leftideal of A . Suppose that A/R is separable. Then there exists a semisimple subalgebra S of A such that A = S ⊕ rad A and I = I S ⊕ I rad A , where I S = I T S and I rad A = I T rad A . Recall that the rank of a perfect finite dimensional algebra A is the smallest rank ofthe simple components of A/ rad A . Theorem 4.5.
Every simple locally finite associative algebra over F has a perfect localsystem of arbitrary large rank.Proof. Let A be a simple locally finite associative algebra. Then by Theorem 4.1, A hasa perfect local system. Let { A α } α ∈ Γ be a perfect local system of A . Then by Theorem4.3, for each α ∈ Γ , there is γ ∈ Γ such that A α ⊂ A γ and M γ ∩ A α = 0 for somemaximal ideal M γ of A γ . Since A γ is finite dimensional, A γ = S γ ⊕ R γ , where S γ is aLevi subalgebra of A γ and R γ = rad A γ is the radical of A γ . Let { S γ , . . . , S nγ } be the setof the simple components of S γ . First we claim that R γ ⊆ M γ . Assume to the contrarythat R γ * M γ . Then M γ + R γ = M γ . Since M γ is maximal, A γ = M γ + R γ . Thus, A γ /M γ = ( M γ + R γ ) /M γ ∼ = R γ / ( R γ ∩ M γ ) = 0 is a non-zero nilpotent quotient algebra of A γ , but A γ = A γ (as A γ are perfect for all γ ), so A γ = A nγ ⊆ M γ , a contradiction. Thus, R γ ⊂ M γ . Note that M γ + S γ (because M γ = A γ ). Since A γ /M γ = ( S γ ⊕ R γ ) /M γ = ( S γ + M γ ) /M γ ∼ = S γ / ( S γ ∩ M γ ) = 0 , IMPLE LOCALLY FINITE ASSOCIATIVE ALGEBRAS 7 A γ /M γ ∼ = S iγ for some ≤ i ≤ n . Since A α ∩ M γ = 0 , we have that A α ⊆ S iγ .Therefore, there is a simple component S iγ in every Levi subalgebra S γ of A γ such that dim S iγ ≥ dim A γ . (cid:3) -Perfect and Conical Local Systems Definition 5.1. [10] Let A be an associative algebra over a field F . Then A is called - perfect if A has no ideals of codimension .An ideal I of A is called - perfect if as an algebra I is -perfect. By using the secondand the third Isomorphism Theorems, we obtain the following well known properties. Lemma 5.2. [10] (i) The sum of -perfect ideals of an associative algebra A is a -perfect ideal of A .(ii) Let P be a -perfect ideal of A and let C be a -perfect ideal of A/P . Then thefull preimage of C in A is -perfect. By using Lemma 5.2(i) we get that every associative algebra contains the largest -perfect ideal. Definition 5.3.
Let A be an associative algebra and let P A be the largest -perfectideal of A . Then P A is said to be the -perfect radical of A .We will need the following simple fact. Lemma 5.4.
Let P be an ideal of an associative algebra A over a field F . If P ′ is anideal of P with P ′ = P ′ , then P ′ is an ideal of A . The following results due to Baranov and Shlaka [10] Shows that P A has radical-likeproperties. Proposition 5.5.
Let A be an associative algebra over F . The following hold.(1) P A = P A .(2) P P A = P A .(3) P A/ P A = 0 .(4) Consider any maximal chain of subalgebras A = A ⊃ A ⊃ · · · ⊃ A r of A suchthat A i +1 is an ideal of A i . If dim A i /A i +1 = 1 for all ≤ i ≤ r − , then A r = P A . Theorem 5.6.
Any simple locally finite associative algebra posses a -perfect localsystem.Proof. By Theorem 4.1 A contains a perfect system. Let { A α } α ∈ Γ be a perfect systemof A . Consider the largest -perfect radical P A α of A α for each α ∈ Γ . Since P A α is finite dimensional, P A α = S α ⊕ rad P A α for some Levi subalgebra S α of P A α . Let { S α , . . . , S nα } be the set of the simple components of S α . We denote by A iα to be theideal of P A α generated by S iα . Fix any index, say . Since A α is perfect, by Lemma 5.4, A α is an ideal of A α . Thus, by Proposition 3.5, { A α } α ∈ Γ S α is a -perfect local systemof A , where Γ S α = { β ∈ Γ | A β ⊃ A α } . (cid:3) Definition 5.7.
Let { A α } α ∈ Γ be a perfect local system of A . Then { A α } α ∈ Γ is said tobe conical if Γ contains a minimal element such that HASAN M. S. SHLAKA A ⊆ A α for all α ∈ Γ ;2. A is simple;3. for each α ∈ Γ the restriction of any natural A α -module to A has a non-trivialcomposition factor.We denote by the rank of the conical local system is the rank of the perfect algebra A . Theorem 5.8.
Every simple locally finite associative algebra over F has conical localsystem of arbitrary large rank.Proof. Let A be a simple locally finite associative algebra over F . By Corollary 4.5, A has a perfect local system of arbitrary large rank. Suppose that { A α } α ∈ Γ is a perfectlocal system of A of arbitrary large rank. Fix A β ∈ { A α } α ∈ Γ . Since A β is finitedimensional associative algebra, there is a Levi subalgebra S β of A β such that A β = S β ⊕ R β , where R β = rad A β is the radical of A β . Let S be a simple component of S β . For all γ ≥ β we denote by A sγ to be the ideal of A γ that generated by S . Then A sγ is the smallest ideal of A γ that contains S . Since ( A sγ ) ⊆ A sγ is also an idealof A γ that contains S , we get that ( A sγ ) = A sγ , so A sγ is perfect. Put A s = S and γ s = { γ ∈ γ | γ ≥ β } ∪ { } . Put A s = ∪ γ ∈ γ s A sγ . Since A sγ ⊆ A sγ ′ for all γ and γ ′ ∈ γ s with γ ≤ γ ′ , we get that A s = lim −→ A sγ is an ideal of A . But A is simple, so A s = A and { A sγ } γ ∈ γ s is a perfect local system of A . Moreover, we have { A sγ } γ ∈ γ s is a conical localsystem of A because { A sγ } γ ∈ γ s is a perfect local system with γ s containing a minimalelement that satisfies the conditions (1), (2) and (3) of Definition 5.7, so it is a localsystem of arbitrary large rank, as required. (cid:3) Theorem 5.9.
Every simple locally finite associative algebra over F has -perfect con-ical local system .Proof. Let A be a simple locally finite associative algebra over F . By Theorem 5.6, A has a -perfect local system. It remains to follow the same process as in the proof ofTheorem 5.8, we get the required results. (cid:3) Acknowledgement:
I would like to express my appreciation to my Supervisor Dr Alexander Baranov for hisadvice and encouragement.
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Department of Mathematics, Faculty of Computer Science and Mathematics, Uni-versity Kufa, Al-Najaf, Iraq.
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