the multiplication algebra of a b-semisimple baric algebra

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This article was downloaded by: [University of Windsor] On: 12 November 2014, At: 07:58 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Communications in Algebra Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lagb20 THE MULTIPLICATION ALGEBRA OF A B-SEMISIMPLE BARIC ALGEBRA M. A. Couto a & H. Guzzo Jr. b a Departamento de Matemática, Campus Universitário Lagoa Nova , Universidade Federal do Rio Grande do Norte , Natal, RN, 59072-970, Brazil b Universidade de São Paulo, Instituto de Matemática e Estatística , Caixa Postal 66281, São Paulo, 05315-970, Brazil Published online: 16 Aug 2006. To cite this article: M. A. Couto & H. Guzzo Jr. (2001) THE MULTIPLICATION ALGEBRA OF A B-SEMISIMPLE BARIC ALGEBRA, Communications in Algebra, 29:4, 1729-1740, DOI: 10.1081/AGB-100002129 To link to this article: http://dx.doi.org/10.1081/AGB-100002129 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

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Page 1: THE MULTIPLICATION ALGEBRA OF A B-SEMISIMPLE BARIC ALGEBRA

This article was downloaded by: [University of Windsor]On: 12 November 2014, At: 07:58Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Communications in AlgebraPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lagb20

THE MULTIPLICATION ALGEBRA OF A B-SEMISIMPLEBARIC ALGEBRAM. A. Couto a & H. Guzzo Jr. ba Departamento de Matemática, Campus Universitário Lagoa Nova , Universidade Federaldo Rio Grande do Norte , Natal, RN, 59072-970, Brazilb Universidade de São Paulo, Instituto de Matemática e Estatística , Caixa Postal 66281,São Paulo, 05315-970, BrazilPublished online: 16 Aug 2006.

To cite this article: M. A. Couto & H. Guzzo Jr. (2001) THE MULTIPLICATION ALGEBRA OF A B-SEMISIMPLE BARIC ALGEBRA,Communications in Algebra, 29:4, 1729-1740, DOI: 10.1081/AGB-100002129

To link to this article: http://dx.doi.org/10.1081/AGB-100002129

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: THE MULTIPLICATION ALGEBRA OF A B-SEMISIMPLE BARIC ALGEBRA

COMMUNICATIONS IN ALGEBRA, 29(4), 1729–1740 (2001)

THE MULTIPLICATION ALGEBRA OFA B-SEMISIMPLE BARIC ALGEBRA

M. A. Couto1,∗,† and H. Guzzo, Jr.2,#

1Universidade Federal do Rio Grande do Norte,Departamento de Matematica, Campus Universitario Lagoa

Nova 59072-970, Natal - RN, Brazil2Universidade de Sao Paulo, Instituto de Matematica e

Estatıstica, Caixa Postal 66281, 05315-970,Sao Paulo, Brazil

ABSTRACT

In this paper we study the multiplication algebra M(A) of an arbi-trary baric algebra A of finite dimension with idempotent elementof weight 1. First, we prove that if A is a baric algebra, then its mul-tiplication algebra is baric too. We give an example where the baricalgebra A is b-semisimple, that is, in the baric sense, but its mul-tiplication algebra M(A) is not b-semisimple. Then, we provideconditions under which this multiplication algebra is b-semisimple,when the baric algebra A is b-semisimple. We finish with a partialconverse, that is, if M(A) is b-semisimple and A2 = A, then A isb-semisimple.

1991 Mathematics Subject Classification: 17D92.

∗The author was sponsored by CAPES-PICD.†E-mail: [email protected]#E mail: [email protected]

1729

Copyright C© 2001 by Marcel Dekker, Inc. www.dekker.com

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1. BARIC ALGEBRAS

Baric algebras play a central role in the theory of genetic algebras. Theywere introduced by I.M.H. Etherington, [5], aiming for an algebraic treatment ofGenetic Populations. Several classes of baric algebras have been defined: train,Bernstein, special triangular, etc.

Let F be a field and A an algebra over F , not necessarily associative,commutative or finite dimensional. If ω :A → F is a nonzero homomorphism,then the ordered pair (A, ω) will be called a baric algebra over F and ω itsweight function. For x ∈ A, ω(x) is called weight of x . If B ⊆ A, we will de-note bar (B) = {x ∈ B : ω(x) = 0}. We observe that bar (B) = B if, and only if,B ⊆ bar (A).

When B is a subalgebra of A and B ⊆ bar (A), then B is called baric subal-gebra of (A, ω). In this case (B, ω′) is a baric algebra, where ω′ = ω | B :B → F .If B is a baric subalgebra of (A, ω) and bar (B) is a two-sided ideal of bar (A)(then by [7, Prop. 1.1] it is also a two-sided ideal of A), then B is called normalbaric subalgebra of (A, ω).

Let B be a baric subalgebra of (A, ω). Then bar (B) is a two-sided ideal ofB of codimension 1, called the bar ideal of B. For all b ∈ B with ω(b) = 0, wehave B = Fb ⊕ bar (B). If I ⊆ bar (B) is a two-sided ideal of B, then I is called abaric ideal of B. If I is a baric ideal of A and I does not contain any proper idealsof A, then I is called a simple baric ideal of A.

Definition 1.1. Let (A, ω) be a baric algebra and B ⊆ A.

(i) B is a (normal) maximal baric subalgebra of (A, ω) if B is a (normal)baric subalgebra of A and there is no C, (normal) baric subalgebra ofA, such that B ⊂ C ⊂ A.

(ii) B is a maximal baric ideal of (A, ω), ifB is a baric ideal of (A, ω), B =bar (A) and there is no J, baric ideal of (A, ω), such that B ⊂ J ⊂bar (A).

Definition 1.2. A baric algebra (A, ω) is simple if for all B, either bar (B) = 0or bar (B) = bar (A), where B is a normal baric subalgebra of A.

Definition 1.3. Let (A, ω) be a baric algebra. The bar-radical of (A, ω) is zero if(A, ω) is simple, otherwise the bar-radical of (A, ω) is the intersection of bar (B),where B, is a normal maximal baric subalgebra of A. We will denote RB(A) thebar-radical of a baric algebra (A, ω).

Clearly RB(A) is a baric ideal of (A, ω).

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Proposition 1.1. Let (A, w) be a baric algebra with an idempotent of weight 1.Then the following conditions hold:

(i) RB(A) is the intersection of bar(B), where B runs over the set of allmaximal normal baric subalgebra of A.

(ii) RB(A) is the intersection of I , where I runs over the set of all maximalbaric ideals of A.

Proposition 1.2. LetB be a baric normal subalgebra of the baric algebra (A, w)such that RB(A/bar (B)) = 0. Then, RB(B) ⊆ RB(A) ⊆ B.

The proof can be seen in [8, Prop. 3.4].

Definition 1.4. We say that (A, ω) is a b-semisimple baric algebra if A has anidempotent element e ∈ A of weight 1 and A is written as

A = Fe ⊕ I1 ⊕ . . . ⊕ Ik

where I j , for j = 1, . . . , k are simple baric ideals of A.

If A has finite dimension and idempotent of weight 1, we have that A isb-semisimple if and only if RB(A) = 0 (see [8]).

Theorem 1.1. If A is a baric associative algebra of finite dimension, then A hasan idempotent element of weight 1.

The proof can be see in [4]

2. THE FIRST RESULT

We remark that throughout this paper, we will work with finite dimensionallgebras.

Let a be any element of an algebraA over F. The right multiplication Ra andthe left multiplication La ofA, which are determined by a, are defined respectivelyby

Ra : x → xa, La : x → ax for all x ∈ A

wich are linear operators on A. The set R(A) (and L(A)) of all right (and left)multiplications of A is a subspace of the associative algebra of all linear operatorson A. We denote by M(A), the enveloping algebra of R(A) ∪ L(A); that is, the

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associative subalgebra of the associative algebra of all linear operators on A, gen-erated by right and left multiplications of A. We have that M(A) is the set of allpolynomials f (Sa1 , Sa2 , . . . , San ), n ≥ 1, f ∈ F{X1, . . . , Xn}, with null indepen-dent term and Sai = Lai or Rai , where F{X1, . . . , Xn} is the set of all associativeand noncommutative polynomials over F .

It is sometimes useful to have a notation for the enveloping algebra of theright and left multiplications which corresponds to the elements of any subset B ofA; we shall write [B]∗ for this subalgebra of M(A). That is, [B]∗ is the set of allf (Sb1 , . . . , Sbk ), where Sbi is either Rbi , the right multiplication of A determinedby bi ∈ B, or Lbi .

Nathan Jacobson in [9], proved that if an arbitrary algebra A is written asa direct sum of simple subalgebras, then its multiplication algebra M(A) also iswritten as a direct sum of simple subalgebras.

We would like to prove that when the baric algebra A is b-semisimple, itsmultiplication algebra M(A) is too. But this is not true and we will show it withan example. Before doing this, we need to know if the multiplication algebra isbaric. We have the following theorem:

Theorem 2.1. If (A, w) is a baric algebra, then M(A) is a baric algebra.

Proof: As bar (A) is an ideal of codimension 1 and bar (A) + A2 = A, so thatA/bar (A) is an irreducible module for the multiplication algebra M(A), andthis amounts to the existence of the correponding representation, which givesa nonzero homomorphism λ :M(A) → F (∼= EndF (A/bar (A))). Notice that ife ∈ A with ω(e) = 1, then e 2 = e ∈ A/bar (A) and for any f ∈ M(A), if f (e) −αe ∈ bar (A), then λ( f ) = α by its own definition. Hence λ( f ) = ω( f (e)).

By Theorems 1.1 and 2.1 we have,

Corollary 2.1. If (A, ω) is a finite dimensional baric algebra, then M(A) hasan idempotent element of weight 1.

3. THE MULTIPLICATION ALGEBRA OF A B-SEMISIMPLEALGEBRA

In this section all baric algebras will be assumed to contain an idempotent ofweight 1. Let A be a b-semisimple algebra and M(A) its multiplication algebra. IfR(M) is the nil radical of M(A), that is, the set of the properly nilpotent elementsof M(A), we have the following characterization for this radical.

Theorem 3.1. Let (A, w) be a b-semisimple algebra. If R(M) is the nil radicalof (M(A), λ) then, R(M) = { f ∈ bar (M(A)) | f|bar (A) ≡ 0}.

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B-SEMISIMPLE BARIC ALGEBRA 1733

Proof: As (A, w) is b-semisimple then there are I1, . . . , Ik minimal ideals of Asuch that bar (A) = I1 ⊕ · · · ⊕ Ik . Therefore A/I and I1, . . . , Ik are irreduciblemodules for M(A),

R(M)(A/bar (A)) = 0 and R(M)Ii = 0 (i = 1, . . . , k), (1)

soR(M)A ⊆ bar (A) andR(M)bar (A) = 0 and it follows thatR(M)2 = 0. ButifJ = ann(A/bar (A)) ∩ ann(bar (A)) = { f ∈ M(A) : f A⊆ bar (A), f bar (A)= 0}, then R(M) ⊆ J by (1) and J 2 = 0 by the same argument above. ThusJ ⊆ R(M). Therefore the Jacobson radical of M(A) is

R(M) = ann(A/bar (A)) ∩ ann(bar (A)),

besides, ann (A/bar (A)) = bar (M(A)) by the arguments in the proof of Theo-rem 2.1, so the result follows.

Corollary 3.1. R(M)2 ≡ 0.

Let e be an idempotent of (A, ω) of weight 1 and let [Fe]∗ be the subalgebraof M(A) generated by Le, Re. This is a baric subalgebra, because Re, Le ∈[Fe]∗ and λ(Re) = λ(Le) = 1. We can decompose this subalgebra in the followingform:

[Fe]∗ = FLe ⊕ bar ([Fe]∗).

Notice that, if f ∈ bar ([Fe]∗), then f|Fe ≡ 0.

Now, let [I j ]∗ be the subalgebras of M(A) generated by the ideals I j , forj = 1, . . . , k. From these algebras, we consider the ideals of M(A) generated bythem. We denote these ideals by U j . We observe that these ideals can be null.For example: if the ideals I j are composed of absolute divisors of zero. So, in thefollowing results we will be considering the nontrivial ideals U j .

Proposition 3.1. The elements of U j , for j = 1, . . . , k, are f j (Sa1 , . . . , Sam ),where Sai = Lai or Sai = Rai , for i = 1, . . . , m, with ai ∈ I j or ai = e, and ineach monomial of f j there exists a Sai with ai ∈ I j .

Proof: First, we note by construction that [I j ]∗ is generated by I j , and the ele-ment e is not in I j . Then, necessarily there exists in each monomial a multiplicativeoperator defined by an element of bar (A). Now, we will show that this element ofbar (A) is in I j . For this, we consider f j (Sa1 , . . . , Sam ) ∈ U j . We write f j as

f j(Sa1 , . . . , Sam

) =n∑

i=1

αi Sai1. . . Saik

.

We suppose that there exists ail , 1 ≤ i ≤ n, 1 ≤ l ≤ m, such that ail ∈ Is, s = jand Sail

is present in monomials of f j . If αi Sai1. . . Sail

. . . Saikis any of these

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1734 COUTO AND GUZZO

monomials, it follows that αi Sai1. . . Sail

. . . Saik(x) ∈ I j ∩ Is = (0), for all x ∈ A.

So, we showed that all these monomials are null and the proposition follows.

Corollary 3.2. If i = j, then Ui ∩ U j = (0).

Corollary 3.3. If f j ∈ U j , it follows that f j (ai ) = 0, for ai ∈ Ii and i = j.

Corollary 3.4. We have that M(A) is written as

M(A) = (FLe ⊕ bar ([Fe]∗)) + (U1 ⊕ U2 ⊕ · · · ⊕ Uk).

Proposition 3.2. The intersection R(M) ∩ U j is either 0 or a simple baric idealof M(A).

Proof: Since it is the intersection of two baric ideals, it is a baric ideal.By Theorem 3.1 and Corollary 3.3, R j (M) = { f ∈ U j : f (I j ) = 0} and

R j (M)[A] = R j (M)[e] ⊆ I j . Since I j is a minimal ideal (an irreducible M(A)-submodule) and the action ofM(A) onA is faithful,R j (M)[e] = I j and the linearmap R j (M) → I j : f j �→ f j (e) is an isomorphism of M(A)-modules. ThereforeR j (M) is an irreducible left ideal and, a fortiori, a simple ideal.

Corollary 3.5. Under the conditions of the Proposition 3.2, if R j (M) = 0 wehave that R j (M) has the same dimension of I j .

Proposition 3.3. If R(M) ∩ U j = U j = 0, then I j = U j (e) is nilpotent ofindex two.

Proof: If a ∈ I j , then Ra, La ∈ U j . Hence, Ra(x) = 0, La(x) = 0, for all x ∈I j .So, I 2

j = (0).

Proposition 3.4. If (0) = U j ∩ R(M) = R j (M) = U j , then, U j = R j (M) ⊕Jj , a direct sum of M(A)−submodules, and Jj = { f ∈ U j | f|Fe ≡ 0}.

Proof: It is enough to note that the map U j → I j : f �→ f (e) is a surjective ho-momorphism ofM(A)-modules with kernel Jj , which restricts to the isomorphismR j (M) → I j considered in Proposition 3.2.

Proposition 3.5. If U j ∩ R(M) = (0), then U j is a simple baric ideal.

Proof: If U j = 0, since U j ⊆ ann(A/bar (A)) ∩ (∩i = j ann(Ii )), U j ∩ ann(I j )= 0, so that I j is an irreducible and faithful module for U j , forcing U j to be simple.

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The ideals U j , of the proposition above are associative algebras. Also, sincethey are simple ideals, they are simple subalgebras or they are null.

Proposition 3.6. The subspace Jj of the Proposition 3.4 has a function f j suchthat f j ≡ I d|I j .

Proof: As in the proof of Proposition 3.5, I j is an irreducible and faithful modulefor Jj , hence Jj is a simple algebra and it is enough to take f j the unit element ofJj .

Proposition 3.7. If U j ∩ R(M) = U j , for every j = 1, . . . , k, then M(A) iswritten as follows, M(A) = F Le ⊕ U1 ⊕ U2 ⊕ · · · ⊕ Uk .

Proof: To prove this proposition, we have to show that the elements of bar ([Fe]∗)are written as a linear combination of elements of the ideals U j , for j = 1, . . . , k.

We observe that according with the proofs of the Proposition 3.5 and Propo-sition 3.6, there exist functions f j in U j such that f j |I j

= I d |I j . Now, if f ∈bar ([Fe]∗), then f j f has the same image of the function f|I j . In fact, as weobserved, f (e) = 0 and then f j f (e) = 0. Also, if a ∈ bar (A), we write a asa = a1 + a2 + · · · + ak with a j ∈ I j for j = 1, . . . , k. Then,

f j f (a) = f j f (a1 + a2 + · · · + ak) = f j f (a j ) = f (a j ).

Since U j is an ideal, it follows that f j f ∈ U j and so, we write f = f1 f + · · · +fk f, that is, f is written as a linear combination of elements of the ideals U j . So, theproposition follows.

Now, we suppose that U j ∩ R(M) = U j for 1 ≤ j ≤ m with m < k. Underthis assumption, the elements of bar ([Fe]∗) cannot be written as linear combina-tion of elements in U j , but, there exists a decomposition for the algebra M(A).

Since the element h = fm+1 + · · · + fk is an idempotent which commuteswith [Fe]∗ + U1 + · · · + Um, B = (1 − h)bar ([Fe]∗) is a subalgebra of M(A).

Now, there exists f ∈ bar ([Fe]∗), for all g ∈ B, such that g is equal tof|I1⊕···⊕Im . In these conditions, we have the following proposition:

Proposition 3.8. If U j ∩ R(M) = U j for 1 ≤ j ≤ m and U j ⊂ R(M) for m +1 ≤ j ≤ k with m < k, then M(A) is written as follows:

M(A) = F Le ⊕ B ⊕ U1 ⊕ U2 ⊕ · · · ⊕ Uk .

Proof: Notice that B = { f − fm+1 f − · · · − fk f | f ∈ bar ([Fe]∗)}.Given h ∈ B ∩ (U1 + · · · + Uk), then h = h1 + · · · + hk = f − fm+1 f −

· · · − fk f , where hi ∈ Ui , f ∈ bar ([Fe]∗).

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Since h ∈ B then h(e) = 0. But hi (e) ∈ Ii so hi (e) = 0.Given a ∈ bar (A) then a = a1 + · · · + ak , where ai ∈ Ii . For i = 1, . . . , m,

we have hi ∈ Ui ⊆ R(M) then h(a) = hm+1(a) + · · · + hk(a) and h j (ai ) = 0 forj = m + 1, . . . , k.

For i = m + 1, . . . , k we have h(ai ) = f (ai ) − fm+1 f (ai ) − · · · − fk f (ai )= f (ai ) − fi f (ai ) = f (ai ) − f (ai ) = 0.

Corollary 3.6. If U j ∩ R(M) = U j for j = 1, . . . , k then, M(A) is written as

M(A) = F Le ⊕ bar ([Fe]∗) ⊕ U1 ⊕ · · · ⊕ Uk

Proposition 3.9. The radicalR(M) ofM(A) is contained in the direct sum U1 ⊕· · · ⊕ Uk and R(M) ∩ (U1 ⊕ · · · ⊕ Uk) = U1 ∩ R(M) ⊕ · · · ⊕ Uk ∩ R(M).

Proof: By Proposition 3.8, any element f ∈ R(M) decomposes as f = f0 +f1 + · · · + fk with f0 ∈ B and fi ∈ Ui for i = 1, . . . , k. Then for j = 1, . . . ,

k, 0 = f (I j ) = f0(I j ) since I 2j = 0, so f0(e) = 0, f0(I j ) = 0 for any j and f0 =

0. Hence f ∈ U1 ⊕ · · · ⊕ Uk . Now R(M) ∩ (U1 ⊕ · · · ⊕ Uk) = U1 ∩ R(M) ⊕· · · ⊕ Uk ∩ R(M) by wellknown facts of associative algebras.

To finish, using Propositions 3.5, 3.7, 3.8 and 3.9, we have that of multi-plication algebra M(A) of a b-semisimple algebra, can be written as M(A) =R(M) ⊕ G, where G is a subalgebra that can assume one of the following de-compositions:

(i) G = F Le ⊕ J1 ⊕ · · · ⊕ Js ⊕ Us+1 ⊕ · · · ⊕ Uk, for m = 0,where Ul = Jl ⊕ Rl(M) for l = 1, . . . , s,

(ii) G = F Le ⊕ B ⊕ Jm+1 ⊕ · · · ⊕ Js ⊕ Us+1 ⊕ Uk for m = 0,where B = { f − fm+1 f − · · · − fk f | f ∈ bar ([Fe]∗)}.

(iii) G = F Le ⊕ bar [Fe]∗ where U j ∩ R(M) = U j for j = 1, . . . , k.

Now, we will give necessary and sufficient conditions so that a multiplica-tion algebra of a b-semisimple algebra, is b-semisimple. We have the followingtheorem:

Theorem 3.2. Let (A, w) be a b-semisimple algebra. Then M(A) =F Le ⊕ B ⊕ R1(M) ⊕ · · · ⊕ Rm(M) ⊕ · · · ⊕ Rs(M) ⊕ Jm+1 ⊕ · · · ⊕ Js

⊕Us+1 ⊕ · · · ⊕ Uk,

is b-semisimple if, and only if, J j ≡ 0 for j = m + 1, . . . , s and B ≡ 0.

Proof: According [4, Th. 4.2], we have that RB(M) = R(M) ∩ (bar (M))2.

Then, the radical isRB(M) = BR1(M) ⊕ · · · ⊕ BRm(M) ⊕ J1R1(M) ⊕ · · · ⊕JsRs(M), because (Ri (M))2 ≡ 0 for i = 1, 2, . . . , m.

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In order toM(A) be b-semisimple is necessary and sufficient that the radicalabove is null. Hence BRi (M) ≡ 0, for i = 1, 2, . . . , m and Jj R j ≡ 0, for j =1, 2, . . . , s. We know that the functions of B can be different of zero only inthe ideals Ii for i = 1, 2, . . . , m. So, BRi (e) = B(Ii ) ≡ 0, for i = 1, 2, . . . , m.Hence B ≡ 0. Analogously, we have that Jj R j (e) = Jj (I j ) ≡ 0, and then Jj ≡ 0for j = 1, 2, . . . , s, because Jj is different form zero only in I j . So, the resultfollows.

Proposition 3.10. Let (A, ω) be a baric algebra with an idempotent of weight 1.If B = bar ([Fe]∗) and M(A) is b-semisimple, then ea = ae, for all a ∈ bar (A).

Proof: If M(A) is b-semisimple, then BR(M) = (0). So, B[bar (A)] = (0).Therefore, (Le − Re)(a) = 0 for all a ∈ bar (A). Hence ea = ae.

Corollary 3.7. If (A, w) and M(A) are b-semisimple, then eI j = I j e, for allnilpotent baric ideals I j of A, and e is the idempotent of weight 1 of A.

Proof: Note that

B = {f ∈ M(A) : f (e) = 0, f (I j ) = 0 ∀ j > m

and ∃ g ∈ bar ([Fe]∗) with f |I j = g|I j ∀ j ≤ m}.

So, if M(A) is b-semisimple, then B = (0) and (Le − Re)|I j ≡ 0. Therefore, theresult follows.

Example 1. We will give an example of a b-semisimple algebra, such that itsmultiplication algebra is not b-semisimple. Let A be a baric algebra generated bythe elements {e, v}, with the multiplication table:

e v

e e 2v

v 2v 2v

and weight function w : A → F given by w(e) = 1 and w(v) = 0. Note that thisalgebra is commutative but is not associative.

We identify EndF (A) with the algebra Mat2(F) by taking coordinate ma-trices in the given basis. Then the matrix corresponding to Le is ( 1 0

0 2 ) and toLv is ( 0 0

2 2 ). Under this identification M(A) = {( α 0β γ ) : α, β, γ ∈ F}. The bar-

radical of M(A) is bar (M(A)) = {( 0 0β γ ) : β, γ ∈ F} = bar ([Fe]∗), the radical

is R(M) = {( 0 0β 0 ) : β ∈ F} and everything is obvious. Almost no computation is

needed.

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1738 COUTO AND GUZZO

Example 2. We will consider the following subalgebra A of the matrix algebraM2(F)

A ={(

α β

0 0

)∈ M2(F)

}

If we take w : A → F, given by

w

((α β

0 0

))= α,

then (A, w) is a baric algebra. This algebra is b-semisimple, because it can bewritten as

F

(1 0

0 0

)⊕

⟨(0 1

0 0

)⟩F

.

Taking the multiplication algebra M(A), we have that

M(A) = F Re ⊕ F (Re − Le) ⊕ F Rv,

where, e = ( 1 00 0 ), v = ( 0 1

0 0 ).In this algebraB = F(Re − Le) andR(M) = F Rv. Note thatBR(M)[e] =

B[〈v〉] = 〈v〉 and the necessary condition ev = ve is not verified in this algebra.

4. THE B-SEMISIMPLE MULTIPLICATION ALGEBRAS

We consider a baric algebra (A, w) and its multiplication algebra M(A). Anatural question is: if the multiplication algebra is b-semisimple, does it followsthat the algebra A is b-semisimple too? In this direction, we have the followingpropositions:

Proposition 4.1. If A2 = A, then bar (M(A))[A] = bar (A).

Proof: We observe thatM(A) = F Le ⊕ bar (M(A)) and if x ∈ e · bar (A) thenx = ey with y ∈ bar (A), hence x = Ry(e) ∈ bar (M(A))[A]. So,

A = Fe + bar (A) = A2 = M(A)[A] = F Le[A] + bar (M(A))[A]

= Fe + e.bar (A) + bar (M(A))[A] = Fe + bar (M(A))[A].

Therefore, bar (M(A))[A] = bar (A).

We suppose that M(A) is b-semisimple and that A2 = A. Under theseconditions, we have that bar (M(A)) is written as a direct sum of simple baricideals U1 ⊕ · · · ⊕ Uk . If we suppose that this algebra is not semisimple in the

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B-SEMISIMPLE BARIC ALGEBRA 1739

usual sense, according [4, Pr. 4.2], some of these ideals form the nil radical R(M)of M(A), with R(M)2 = (0) and the other ideals, if they exist, have the propertythat U 2

i = Ui . Here we denote U 2i = 0 for i = 1, 2, . . . , m and U 2

i = Ui , for i =m + 1, . . . , k. Under these conditions, we have the following result:

Theorem 4.1. If (A, w) is a baric algebra such that A2 = A and its multiplica-tion algebra M(A) is b-semisimple, then (A, w) is b-semisimple.

Proof: As M(A) is a b-semisimple baric algebra, then bar (M(A)) = U1 ⊕· · · ⊕ Uk where theUi ’s are minimal ideals ordered so thatU 2

i = 0 for i = 1, . . . , mand U 2

i = Ui for i = m + 1, . . . , k. Notice that this implies that Ui is a simple alge-bra for i = m + 1, . . . , k. ThenR(M(A)) = U1 ⊕ · · · ⊕ Um andR(M(A))2 = 0.Let u be the unit element of the semisimple algebra Um+1 ⊕ · · · ⊕ Uk . Then, thePeirce decomposition relative to u givesM(A) = M0 ⊕ M1 (direct sum of ideals)where Mi = {x ∈ M(A) : xu = ux = i x} and, hence M1 = Um+1 ⊕ · · · ⊕ Uk

and U1 ⊕ · · · ⊕ Um ⊆ M0.By lifting the nonzero idempotent in M0/bar (M(A)) ∩ M0 one obtains

that M0 = F f ⊕ U1 ⊕ · · · ⊕ Um , where f 2 = f and, by minimality of the Ui ’s,dim Ui = 1 for any i = 1, . . . , m.

As A is a faithful M(A)-module with M(A)A = A and the codimensionof the submodule bar (A) = bar (M(A))[A] is 1, then A = A0 ⊕ A1, where A1

is a unital M1-module annihilated by M0 and A0 is a M0-module annihilatedby M1. Besides, A1 is a unital module for the (classically) semisimple algebraM1 so A1 = ⊕k

j=m+1U jA1 and each U jA1 is a direct sum of irreducible modulesfor U j , and hence for M1. Also, A1 = M1A1 ⊆ bar (M(A))[A] ⊆ bar (A), sobar (A) = (A0 ∩ bar (A)) ⊕ A1 and A0 ∩ bar (A) = (U1 ⊕ · · · ⊕ Um)A0, since(U1 ⊕ · · · ⊕ Um)bar (M(A)) = 0, (U1 ⊕ · · · ⊕ Um)bar (A) = 0. Finally, A0 =Fe ⊕ (A0 ∩ bar (A)) for some e ∈ A0 so, since A is faithful, the linear map U1 ⊕· · · ⊕ Um → A0 ∩ bar (A), x �→ xe is an isomorphism of M(A)-modules and,therefore, A0 ∩ bar (A) = U1e ⊕ · · · ⊕ Ume is a direct sum of one-dimensionalsubmodules.

ACKNOWLEDGMENT

The authors thank the referee for his comments and suggestions that im-proved the presentation of this paper.

REFERENCES

1. Albert, V.M.: Introduction to Algebra, Proc. Edin. Math. Soc. (2) 20 (1976)53–58.

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2. Costa, R. and Guzzo Jr, H.: Indecomposable baric algebras, Linear Algebraand its Appl. 183: 223–236 (1993).

3. Costa, R. and Guzzo Jr, H.: Indecomposable baric algebras, II, Linear Alge-bra Appl. 196 (1994), 233–242.

4. Couto, M.A. and Guzzo Jr, H.: The radical in alternative baric algebras,Arch. Math. (to appear).

5. Etherington, I.M.H.: On non-associative combinations, Proc. Roy. Soc. Ed-inb. 59 (1939) 153–162.

6. Etherington, I.M.H.: Genetic Algebras, Proc. Roy. Soc. Edinb. 59 (1939)242–258.

7. Guzzo Jr, H.: On normal and composition series for baric algebras, NovaJournal of Mathematics, Game Theory, and Algebra 4(1), 25–38 (1995).

8. Guzzo Jr, H.: The bar-radical of baric algebras, Arch. Math. 67, vol. 2,106–118 (1996). Ibid 16 (1970) 120.

9. Jacobson, N.: A note on nonassociative algebras, Duke Math. J. (3) (544–548) (1937).

10. Schafer, R.D.: Structure of genetic algebras, Amer. J. Math. 71 (1949) 121–153.

11. Schafer. R.D.: An introduction to nonassociative algebras, Academic Press,New York, (1996).

12. Zhevlakov K.A., Slin’ko, A.M., Shestakov, I.P., Shirshov, A.I: Rings that arenearly associative, Academic Press - New York - London - (1982).

Received April 1999Revised January 2000 and June 2000

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