A First Course in Noncommutative Rings

Lam, T. Y.

doi:10.1007/978-1-4419-8616-0

Cited by 768 publications

(647 citation statements)

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“…Remember that a ring is semilocal if its quotient by its Jacobson radical is semisimple artinian (see, for example [5], for more properties about noncommutative rings).…”

Section: The Ringmentioning

confidence: 99%

On the arithmetic of the endomorphisms ring $${{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}$$

Climent

Navarro

Tortosa

2011

AAECC

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For a prime number p, Bergman (1974) established that End(Z p × Z p 2 ) is a semilocal ring with p 5 elements that cannot be embedded in matrices over any commutative ring. We identify the elements of End(Z p × Z p 2 ) with elements in a new set, denoted by E p , of matrices of size 2 × 2, whose elements in the rst row belong to Z p and the elements in the second row belong to Z p 2 ; also, using the arithmetic in Z p and Z p 2 , we introduce the arithmetic in that ring and prove that the ring End(Z p × Z p 2 ) is isomorphic to the ring E p . Finally, we present a Di e-Hellman key interchange protocol using some polynomial functions over E p de ned by polynomial in Z [X].

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“…Remember that a ring is semilocal if its quotient by its Jacobson radical is semisimple artinian (see, for example [5], for more properties about noncommutative rings).…”

Section: The Ringmentioning

confidence: 99%

On the arithmetic of the endomorphisms ring $${{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}$$

Climent

Navarro

Tortosa

2011

AAECC

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“…It is well known (see e.g. [9]) that in the skew field of quaternions, the equivalence class of a quaternion s is characterized by a quadratic equation. A first, yet interesting, result is that the same fact holds also in a (non division) Clifford algebra R n if s ∈ R n+1 and one looks for solutions in R 0 n ⊕ R 1 n :…”

Section: Some Polynomial Equations and Seriesmentioning

confidence: 99%

Slice monogenic functions

2009

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In this paper we offer a new definition of monogenicity for functions defined on R n+1 with values in the Clifford algebra R n following an idea inspired by the recent papers [6], [7]. This new class of monogenic functions contains the polynomials (and, more in general, power series) with coefficients in the Clifford algebra R n . We will prove a Cauchy integral formula as well as some of its consequences. Finally, we deal with the zeroes of some polynomials and power series.

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“…If A/I is left artinian, so is T . Since the finite dimensional subalgebra C * ⊗ 1 centralizers 1 ⊗ A/I, the ideal J = C * ⊗ Jac(A/I) of T is contained in the Jacobson radical of T [15,Prop. 5.7].…”

Section: Semilocal Factor Algebras Of Module Algebrasmentioning

confidence: 99%

Projectivity of Hopf algebras over subalgebras with semilocal central localizations

Skryabin¹

2008

J K-Theor

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The purpose of this paper is to extend the class of pairs A, H where H is a Hopf algebra over a field and A a right coideal subalgebra for which H is proved to be either projective or flat as an A-module. The projectivity is obtained under the assumption that H is residually finite dimensional, A has semilocal localizations with respect to a central subring, and there exists a Hopf subalgebra B of H such that the antipode of B is bijective and B is a finitely generated A-module. The flatness of H over A is shown to hold when H is a directed union of residually finite dimensional Hopf subalgebras, and there exists a Hopf subalgebra of H whose center contains A. More general projectivity and flatness results are established for (co)equivariant modules over an H-(co)module algebra under similar assumptions.

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A First Course in Noncommutative Rings

Cited by 768 publications

References 0 publications

On the arithmetic of the endomorphisms ring $${{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}$$

On the arithmetic of the endomorphisms ring $${{\rm End}(\mathbb{Z}_{p} \times \mathbb{Z}_{p^{2}})}$$

Slice monogenic functions

Projectivity of Hopf algebras over subalgebras with semilocal central localizations

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