The bar-radical of baric algebras

Guzzo, Henrique

doi:10.1007/bf01268924

Cited by 13 publications

(11 citation statements)

References 8 publications

(6 reference statements)

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“…Let us take the quotient baric algebra U/rad(U ). By [5, Corollary 3.1], we have rad(U/rad(U )) = 0, which implies that U/rad(U ) is b-semi-simple, by [5,Theorem 4.2]. So bar(U/rad(U )) is a sum of minimal b-ideals…”

Section: The Bar-radicalmentioning

confidence: 93%

On the Bar-radical of Jordan Baric Algebras

Ferreira

Guzzo

2007

Result. Math.

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Section: The Bar-radicalmentioning

confidence: 93%

On the Bar-radical of Jordan Baric Algebras

Ferreira

Guzzo

2007

Result. Math.

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“…According to [6], a baric algebra ( A, ω) is b-simple if for all normal baric subalgebras B of ( A, ω) either bar(B) = 0 or bar(B) = bar( A). Also by [6], the bar-radical of ( A, ω) is zero if ( A, ω) is b-simple, otherwise the bar-radical of ( A, ω) is the intersection of all bar(B), where B is a maximal normal baric subalgebra of ( A, ω).…”

Section: The Bar-radical In N Th -Order Bernstein Algebrasmentioning

confidence: 99%

“…Also by [6], the bar-radical of ( A, ω) is zero if ( A, ω) is b-simple, otherwise the bar-radical of ( A, ω) is the intersection of all bar(B), where B is a maximal normal baric subalgebra of ( A, ω). We will denote by rad( A) the bar-radical of a baric algebra ( A, ω).…”

Section: The Bar-radical In N Th -Order Bernstein Algebrasmentioning

confidence: 99%

“…P r o o f. By proof of Theorem 4.3 we have (bar( A)) 2 ֤ J for all J maximal baric ideal of A such that Z 0 ֤ J.Let J be a maximal baric ideal of A such that Z 0 ֤ J,then Z 0 + J = bar( A). For all x ∈ bar(A) we have x = a + b, where a ∈ Z 0 and b ∈ J, then x 2 = 2ab + b 2 ∈ J so (bar( A)) 2 ֤ J.Hence (bar( A)) 2 ֤ rad( A) and by Corollary 4.1 (bar( A)) 2 = rad( A).According to[6] the baric algebra ( A, ω) with an idempotent element is b-semisimple if there exist b-simple baric algebras( A 1 , ω 1 ), . .…”

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The bar-radical in power-associative nth-order Bernstein algebras

Guzzo¹,

Vicente²

2002

Arch. Math.

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“…Hence, according to [7,Prop. 3.4] it follows that r b P 7 r b a 7 barP raX So, r b a 7 ra, and r b a is always nilpotent.…”

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The radical in alternative baric algebras

COUTO

Guzzo

2000

Archiv der Mathematik

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In this paper we proved that in a baric, power associative, F-algebra of finite dimension, there is an idempotent element of weight 1. When this algebra is alternative, with this idempotent we make the Peirces decomposition. We define the nil radical as the maximal nil ideal. We proved that the bar-radical is the intersection of the nil radical with the square of bara. All the results are naturally true for baric associative algebras of finite dimension. 1. Baric algebras. Baric algebras play a central role in the theory of genetic algebras. They were introduced by I. M. H. Etherington, [4], aiming for an algebraic treatment of Genetic 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 wX a À3 F is a nonzero homomorphism, then the ordered pair aY w will be called a baric algebra over F and w its weight funcion or shortly its weight. For x P aY wx is called the weight of x. If b 7 a, we will denote barb fx P b X wx 0g. We observe that barb b if, and only if, b 7 bara.When b is a subalgebra of a and b 7 j bara, then b is called baric subalgebra of aY w. In this case bY w H is a baric algebra, where w H wj b X b À3 F. If b is a baric subalgebra of aY w and barb is a two-sided ideal of bara (then by [6, Prop. 1.1] it is also a two-sided ideal of a), then b is called normal baric subalgebra of aY w.Let b be a baric subalgebra of aY w. Then barb is a two-sided ideal of b of codimension 1. For all b P b with wb j 0, we have b Fb È barb. If I 7 barb is a two-sided ideal of b, then I is called a baric ideal of b. If I is a baric ideal of a and I does not contain any proper ideals of a, then I is called a simple baric ideal of a.

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The bar-radical of baric algebras

Cited by 13 publications

References 8 publications

On the Bar-radical of Jordan Baric Algebras

On the Bar-radical of Jordan Baric Algebras

The bar-radical in power-associative nth-order Bernstein algebras

The radical in alternative baric algebras

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