COMPLEMENTEDLY DENSE IDEALS: DECOMPOSABLE ALGEBRAS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350023
Author(s):  
J. C. CABELLO ◽  
M. CABRERA ◽  
R. ROURA
Keyword(s):  
Direct Summand ◽  
Associative Algebra ◽  
Direct Product ◽  
Subdirect Product ◽  
Extended Centroid ◽  
Dense Ideal ◽  
Central Closure ◽  
Semiprime Algebra ◽  
Multiplication Algebra

An ideal I of a (non-associative) algebra A is dense if the multiplication algebra of A acts faithfully on I, and is complementedly dense if it is a direct summand of a dense ideal. We prove that every complementedly dense ideal of a semiprime algebra is a semiprime algebra, and determine its central closure and its extended centroid. We also prove that a semiprime algebra is an essential subdirect product of prime algebras if and only if, its extended centroid is a direct product of fields. This result is applied to discuss decomposable algebras with respect to some familiar closures for ideals.

Totally multiplicatively prime algebras

2002 ◽  
Vol 132 (5) ◽  
pp. 1145-1162
Author(s):  
M. Cabrera ◽  
Amir A. Mohammed
Keyword(s):  
Associative Algebras ◽  
Central Closure ◽  
The Stability ◽  
Multiplication Algebra

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

scholarly journals Extended centroids of power series rings

1990 ◽  
Vol 32 (3) ◽  
pp. 371-375 ◽  
Author(s):  
W. S. Martindale ◽  
M. P. Rosen ◽  
J. D. Rosen
Keyword(s):  
Prime Ring ◽  
Polynomial Identity ◽  
Extended Centroid ◽  
Prime Rings ◽  
Generalized Polynomial ◽  
Ring Of Quotients ◽  
Power Series Rings ◽  
Central Simple ◽  
Generalized Polynomial Identity ◽  
Central Closure

Prime rings came into prominence when Posner characterized prime rings satisfying a polynomial identity [9]. The scarcity of invertible central elements made it difficult to generalize results from central simple and primitive algebras to prime rings. For example, we do not automatically have tensor products at our disposal. In [5], the first author introduced the Martindale ring of quotients Q(R) of a prime ring R in his theorem characterizing prime rings satisfying a generalized polynomial identity (GPI). Q(R) is a prime ring containing R whose center C is a field called the extended centroid of R. The central closure of R is the subring RC of Q(R) generated by R and C. RC is a closed prime ring since its extended centroid equals its center C. Hence we have a useful procedure for proving results about an arbitrary prime ring R. We first answer the question for closed prime rings and then apply to R the information obtained from RC. It should be noted that simple rings and free algebras of rank at least 2 are closed prime rings. For these reasons, closed prime rings are natural objects to study.

On Identities with Composition of Generalized Derivations

2017 ◽  
Vol 60 (4) ◽  
pp. 721-735 ◽  
Author(s):  
Münevver Pınar Eroglu ◽  
Nurcan Argaç
Keyword(s):  
Prime Ring ◽  
Generalized Derivation ◽  
Multilinear Polynomial ◽  
Extended Centroid ◽  
Generalized Derivations ◽  
Ring Of Quotients ◽  
Central Closure

AbstractLet R be a prime ring with extended centroid C, Q maximal right ring of quotients of R, RC central closure of R such that dim C(RC) > , ƒ (X1, . . . , Xn) a multilinear polynomial over C that is not central-valued on R, and f (R) the set of all evaluations of the multilinear polynomial f (X1 , . . . , Xn) in R. Suppose that G is a nonzero generalized derivation of R such that G2(u)u ∈ C for all u ∈ ƒ(R).

Extended centroid and central closure of semiprime normed algebras. a first approach

1990 ◽  
Vol 18 (7) ◽  
pp. 2293-2326 ◽  
Author(s):  
M. Cabrera Garcia ◽  
A. Rodriguez Palacios
Keyword(s):  
Extended Centroid ◽  
Central Closure

REVISITING HOCHSCHILD COHOMOLOGY FOR ALGEBRA BUNDLES

2008 ◽  
Vol 07 (06) ◽  
pp. 685-715 ◽  
Author(s):  
B. S. KIRANAGI ◽  
R. RAJENDRA
Keyword(s):  
Associative Algebra ◽  
Semidirect Product ◽  
Hochschild Cohomology ◽  
Transitive Group ◽  
Equivalence Classes ◽  
Multiplication Algebra

Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle has been defined and studied in earlier paper. Here, by using cohomological methods, we establish that an algebra bundle is a semidirect product of its radical bundle and a semisimple subalgebra bundle. Further we define multiplication algebra bundle of an algebra bundle and representation of an algebra bundle. We study special representations of an algebra bundle using Hochschild cohomology of an associative algebra bundle with coefficients in a bimodule bundle. We observe that if a representation of an algebra bundle is special then its obstruction is zero. Further we show that a subgroup H of H2(ξ, N) is faithfully represented as a transitive group of translations operating on the set of those equivalence classes of algebra bundle extensions of ξ which determine a given representation [φ, K].

Totally multiplicatively prime algebras

2002 ◽  
Vol 132 (5) ◽  
pp. 1145-1162 ◽  
Author(s):  
M. Cabrera ◽  
Amir A. Mohammed
Keyword(s):  
Associative Algebras ◽  
Central Closure ◽  
The Stability ◽  
Multiplication Algebra

We introduce the totally multiplicatively prime algebras as those normed algebras for which there exists a positive number K such that K‖F‖‖a‖ ≤ ‖WF,a‖ for all F in M(A) (the multiplication algebra of A) and a in A, where WF,a denotes the operator from M(A) into A defined by WF,a(T) = FT(a) for all T in M(A). These algebras are totally prime and their multiplication algebra is ultraprime. We get the stability of the class of totally multiplicatively prime algebras by taking central closure. We prove that prime H*-algebras are totally multiplicatively prime and that the ℓ1-norm is the only classical norm on the free non-associative algebras for which these are totally multiplicatively prime.

EXTENDED CENTROID AND CENTRAL CLOSURE OF MULTIPLICATIVELY SEMIPRIME ALGEBRAS

2001 ◽  
Vol 29 (3) ◽  
pp. 1215-1233 ◽  
Author(s):  
M. Cabrera ◽  
Amir A. Mohammed
Keyword(s):  
Extended Centroid ◽  
Central Closure

scholarly journals Subgroups of 3-Factor Direct Products

2019 ◽  
Vol 73 (1) ◽  
pp. 19-38
Author(s):  
Daniel Neuen ◽  
Pascal Schweitzer
Keyword(s):  
Explicit Formula ◽  
Direct Product ◽  
Subdirect Product ◽  
Structure Theorem ◽  
Symmetric Groups ◽  
Direct Products ◽  
Special Cases ◽  
Subgroup Lattices ◽  
Subdirect Products

Abstract Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.

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