An Introduction to Noncommutative Noetherian Rings

Goodearl, K. R.; Warfield, R. B.

doi:10.1017/cbo9780511841699

Cited by 553 publications

(664 citation statements)

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“…For the commutative cases i = 1, 2, D i is the familiar field of rational functions, but the non-commutative D 3 is likely less familiar (see Ref. [37] for general discussion). Note that the inverse in D 3 of an element r of R 3 can be expressed in a similar way as for quaternions, as an element of R 3 divided by a real polynomial, that is, by an element |r| 2 of R 2 .…”

Section: B Syzygy Theorem and Relation With K0(r)mentioning

confidence: 99%

“…The Gaussian elimination algorithm can be carried out in D i , and the resulting solutions form a set of n 0 − m n 0 -component vectors with entries in D i , and are linearly independent over D i . Finally, we can multiply each n 0 -component vector by a common denominator of its entries [37] to obtain vectors with entries in R i . These must be linearly independent over R i , because if not then the original vectors in D n0 i would be linearly dependent over both R i and D i .…”

Section: B Syzygy Theorem and Relation With K0(r)mentioning

confidence: 99%

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Compactly supported Wannier functions and algebraicK-theory

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2017

Phys. Rev. B

109

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In a tight-binding lattice model with n orbitals (single-particle states) per site, Wannier functions are n-component vector functions of position that fall off rapidly away from some location, and such that a set of them in some sense span all states in a given energy band or set of bands; compactlysupported Wannier functions are such functions that vanish outside a bounded region. They arise not only in band theory, but also in connection with tensor-network states for non-interacting fermion systems, and for flat-band Hamiltonians with strictly short-range hopping matrix elements. In earlier work, it was proved that for general complex band structures (vector bundles) or general complex Hamiltonians-that is, class A in the ten-fold classification of Hamiltonians and band structures-a set of compactly-supported Wannier functions can span the vector bundle only if the bundle is topologically trivial, in any dimension d of space, even when use of an overcomplete set of such functions is permitted. This implied that, for a free-fermion tensor network state with a nontrivial bundle in class A, any strictly short-range parent Hamiltonian must be gapless. Here, this result is extended to all ten symmetry classes of band structures without additional crystallographic symmetries, with the result that in general the non-trivial bundles that can arise from compactlysupported Wannier-type functions are those that may possess, in each of d directions, the non-trivial winding that can occur in the same symmetry class in one dimension, but nothing else. The results are obtained from a very natural usage of algebraic K-theory, based on a ring of polynomials in e ±ikx , e ±iky , . . . , which occur as entries in the Fourier-transformed Wannier functions.

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Section: B Syzygy Theorem and Relation With K0(r)mentioning

confidence: 99%

Section: B Syzygy Theorem and Relation With K0(r)mentioning

confidence: 99%

Compactly supported Wannier functions and algebraicK-theory

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2017

Phys. Rev. B

109

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“…Then R is clearly a domain (hence Armendariz) and J(R) = pR. But R/J(R) is isomorphic to Mat 2 (Z p ) by the argument in [11,Exercise 2A] Considering Lemma 1.1 (3,5), one may naturally ask whether the converse of Lemma 1.1(5) also holds. But the answer is negative as can be seen by R = U n (A) (n ≥ 2) over a reduced ring A.…”

Section: On Radicals When Factor Rings Are Armendarizmentioning

confidence: 99%

On Jacobson and Nil Radicals Related to Polynomial Rings

Kwak¹,

Lee²,

Özcan³

2016

Journal of the Korean Mathematical Society

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Abstract. This note is concerned with examining nilradicals and Jacobson radicals of polynomial rings when related factor rings are Armendariz. Especially we elaborate upon a well-known structural property of Armendariz rings, bringing into focus the Armendariz property of factor rings by Jacobson radicals. We show that J(R[x]) = J(R) [x] if and only if J(R) is nil when a given ring R is Armendariz, where J(A) means the Jacobson radical of a ring A. A ring will be called feckly Armendariz if the factor ring by the Jacobson radical is an Armendariz ring. It is shown that the polynomial ring over an Armendariz ring is feckly Armendariz, in spite of Armendariz rings being not feckly Armendariz in general. It is also shown that the feckly Armendariz property does not go up to polynomial rings.

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“…Then R is a domain and thus nil-semicommutative. However, for any odd prime integer q, there exists a ring isomorphism R/qR ∼ = Mat 2 (Z q ) by the argument in [8,Exercise 2A]. But Mat 2 (Z q ) is not nil-semicommutative by Example 1.7, and thus R/qR cannot be nil-semicommutative.…”

Section: Introductionmentioning

confidence: 99%

Semicommutative Property on Nilpotent Products

Kim¹,

Kwak²,

Lee³

2014

Journal of the Korean Mathematical Society

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Abstract. The semicommutative property of rings was introduced initially by Bell, and has done important roles in noncommutative ring theory. This concept was generalized to one of nil-semicommutative by Chen. We first study some basic properties of nil-semicommutative rings. We next investigate the structure of Ore extensions when upper nilradicals are σ-rigid δ-ideals, examining the nil-semicommutative ring property of Ore extensions and skew power series rings, where σ is a ring endomorphism and δ is a σ-derivation.

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An Introduction to Noncommutative Noetherian Rings

Cited by 553 publications

References 0 publications

Compactly supported Wannier functions and algebraicK-theory

Compactly supported Wannier functions and algebraicK-theory

On Jacobson and Nil Radicals Related to Polynomial Rings

Semicommutative Property on Nilpotent Products

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