scholarly journals On the Dimension of Modules and Algebras, VI. Comparison of Global and Algebra Dimension

1957 ◽  
Vol 11 ◽  
pp. 61-65 ◽  
Author(s):  
Maurice Auslander
Keyword(s):  
Jacobson Radical ◽  
Minimum Condition ◽  
And Algebra ◽  
Algebra Over A Field

Throughout this paper all rings are assumed to have unit elements. A ring Λ is said to be semi-primary if its Jacobson radical N is nilpotent and Г = Λ/N satisfies the minimum condition. The main objective of this paper isTHEOREM I. Let A be a semi-primary algebra over a field K. Let N be the radical of Λ and Г = Λ/N. IfThendim Λ = gl.dim Λ.

Noetherian Tensor Products

1972 ◽  
Vol 15 (2) ◽  
pp. 235-238
Author(s):  
E. A. Magarian ◽  
J. L. Motto
Keyword(s):  
Tensor Product ◽  
Jacobson Radical ◽  
Tensor Products ◽  
Minimum Condition ◽  
Ideal Structure ◽  
The Ideal

Relatively little is known about the ideal structure of A⊗RA' when A and A' are R-algebras. In [4, p. 460], Curtis and Reiner gave conditions that imply certain tensor products are semi-simple with minimum condition. Herstein considered when the tensor product has zero Jacobson radical in [6, p. 43]. Jacobson [7, p. 114] studied tensor products with no two-sided ideals, and Rosenberg and Zelinsky investigated semi-primary tensor products in [9].All rings considered in this paper are assumed to be commutative with identity. Furthermore, R will always denote a field.

On radical extensions of rings

1967 ◽  
Vol 7 (4) ◽  
pp. 552-554 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Jacobson Radical ◽  
Main Tool ◽  
Independent Interest ◽  
Minimum Condition ◽  
Radical Extension ◽  
Commutativity Theorems

A ring K is a radical extension of a subring B if for each x ∈ K there is aninteger n = n(x) > 0 such that xn ∈ B. In [2] and [3], C. Faith considered radical extensions in connection with commutativity questions, as well as the generation of rings. In this paper additional commutativity theorems are established, and rings with right minimum condition are examined. The main tool is Theorem 1.1 which relates the Jacobson radical of K to that of B, and is of independent interest in itself. The author is indebted to the referee for his helpful suggestions, in particular for the strengthening of Theorem 2.1.

scholarly journals Co-stability of Radicals and Its Applications to PI-Theory

2016 ◽  
Vol 23 (03) ◽  
pp. 481-492 ◽  
Author(s):  
A. S. Gordienko
Keyword(s):  
Hopf Algebra ◽  
Lie Algebras ◽  
Associative Algebra ◽  
Jacobson Radical ◽  
Associative Algebras ◽  
Finite Dimensional ◽  
Comodule Algebra ◽  
Graded Polynomial Identities ◽  
Finite Dimensional Lie Algebras ◽  
Algebra Over A Field

We prove that if A is a finite-dimensional associative H-comodule algebra over a field F for some involutory Hopf algebra H not necessarily finite-dimensional, where either char F = 0 or char F > dim A, then the Jacobson radical J(A) is an H-subcomodule of A. In particular, if A is a finite-dimensional associative algebra over such a field F, graded by any group, then the Jacobson radical J(A) is a graded ideal of A. Analogous results hold for nilpotent and solvable radicals of finite-dimensional Lie algebras over a field of characteristic 0. We use the results obtained to prove the analog of Amitsur's conjecture for graded polynomial identities of finite-dimensional associative algebras over a field of characteristic 0, graded by any group. In addition, we provide a criterion for graded simplicity of an associative algebra in terms of graded codimensions.

scholarly journals Geometry and Algebra of the Deltoid Map

2021 ◽  
Vol 128 (1) ◽  
pp. 25-39
Author(s):  
Joshua P. Bowman
Keyword(s):  
And Algebra

On a strong minimum condition of a fractal variational principle

2021 ◽  
pp. 107199
Author(s):  
Ji-Huan He ◽  
Na Qie ◽  
Chun-hui He ◽  
Tareq Saeed
Keyword(s):  
Variational Principle ◽  
Minimum Condition

scholarly journals Kurosh-Amitsur Right Jacobson Radical of Type 0 for Right Near-Rings

2008 ◽  
Vol 2008 ◽  
pp. 1-6
Author(s):  
Ravi Srinivasa Rao ◽  
K. Siva Prasad ◽  
T. Srinivas
Keyword(s):  
Jacobson Radical ◽  
Paper Properties ◽  
Hereditary Radical ◽  
Constant Part ◽  
The Right

By a near-ring we mean a right near-ring.J0r, the right Jacobson radical of type 0, was introduced for near-rings by the first and second authors. In this paper properties of the radicalJ0rare studied. It is shown thatJ0ris a Kurosh-Amitsur radical (KA-radical) in the variety of all near-ringsR, in which the constant partRcofRis an ideal ofR. So unlike the left Jacobson radicals of types 0 and 1 of near-rings,J0ris a KA-radical in the class of all zero-symmetric near-rings.J0ris nots-hereditary and hence not an ideal-hereditary radical in the class of all zero-symmetric near-rings.

Simultaneous Triangularization of Algebras of Polynomially Compact Operators

1991 ◽  
Vol 34 (2) ◽  
pp. 260-264 ◽  
Author(s):  
M. Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Jacobson Radical ◽  
Compact Operators ◽  
General Algebra ◽  
Closed Algebra ◽  
Quasinilpotent Operators ◽  
Simultaneous Triangularization

AbstractIf A is a norm closed algebra of compact operators on a Hilbert space and if its Jacobson radical J(A) consists of all quasinilpotent operators in A then A/ J(A) is commutative. The result is not valid for a general algebra of polynomially compact operators.

Aunt Bertha's Ice Cream Shoppe

2015 ◽  
Vol 108 (9) ◽  
pp. 720
Author(s):  
Katie A. Hendrickson ◽  
Gabrielle Kisner
Keyword(s):  
Problem Based Learning ◽  
Ice Cream ◽  
And Algebra

Problem-based learning motivates students to use their knowledge of geometry and algebra.

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