Direct-sum behavior of modules over one-dimensional rings

2010 ◽  
pp. 251-275 ◽  
Author(s):  
Ryan Karr ◽  
Roger Wiegand
Keyword(s):  
One Dimensional ◽  
Direct Sum

scholarly journals A decomposition theorem for homogeneous algebras

2002 ◽  
Vol 72 (1) ◽  
pp. 47-56 ◽  
Author(s):  
L. G. Sweet ◽  
J. A. Macdougall
Keyword(s):  
Nonzero Element ◽  
Decomposition Theorem ◽  
Small Dimension ◽  
Infinite Field ◽  
Left Multiplication ◽  
One Dimensional ◽  
Direct Sum ◽  
Homogeneous Algebras ◽  
The One ◽  
Homogeneous Algebra

AbstractAn algebra A is homogeneous if the automorphism group of A acts transitively on the one dimensional subspaces of A. Suppose A is a homogeneous algebra over an infinite field k. Let La denote left multiplication by any nonzero element a ∈ A. Several results are proved concerning the structure of A in terms of La. In particular, it is shown that A decomposes as the direct sum A = ker La Im La. These results are then successfully applied to the problem of classifying the infinite homogeneous algebras of small dimension.

scholarly journals Inverse resonance scattering on rotationally symmetric manifolds

2020 ◽  
pp. 1-17
Author(s):  
Hiroshi Isozaki ◽  
Evgeny Korotyaev
Keyword(s):  
Hilbert Spaces ◽  
Counting Function ◽  
Resonance Scattering ◽  
Compact Interval ◽  
Large Radius ◽  
One Dimensional ◽  
Symmetric Manifold ◽  
Direct Sum ◽  
Rotationally Symmetric ◽  
Real Analytic

We discuss inverse resonance scattering for the Laplacian on a rotationally symmetric manifold M = ( 0 , ∞ ) × Y whose rotation radius is constant outside some compact interval. The Laplacian on M is unitarily equivalent to a direct sum of one-dimensional Schrödinger operators with compactly supported potentials on the half-line. We prove Asymptotics of counting function of resonances at large radius. The rotation radius is uniquely determined by its eigenvalues and resonances. There exists an algorithm to recover the rotation radius from its eigenvalues and resonances. The proof is based on some non-linear real analytic isomorphism between two Hilbert spaces.

PURE BRAIDS AS AUTOMORPHISMS OF FREE GROUPS

2005 ◽  
Vol 04 (04) ◽  
pp. 435-440
Author(s):  
MOHAMMAD N. ABDULRAHIM
Keyword(s):  
Braid Group ◽  
Free Groups ◽  
Linear Representation ◽  
One Dimensional ◽  
Pure Braid Group ◽  
Direct Sum ◽  
Automorphisms Of Free Groups ◽  
Diagonal Representation ◽  
Composition Factors

We study the composition of F. R. Cohen's map Pn → Pnk with the Gassner representation, where Pn is the pure braid group. This gives us a linear representation of Pn whose composition factors are one copy of the Gassner representation of Pn and k - 1 copies of a diagonal representation, hence a direct sum of one-dimensional representations.

scholarly journals Direct sum cancellation for modules over one-dimensional rings

2005 ◽  
Vol 283 (1) ◽  
pp. 93-124 ◽  
Author(s):  
Wolfgang Hassler ◽  
Roger Wiegand
Keyword(s):  
One Dimensional ◽  
Direct Sum

scholarly journals Bounded complexes of permutation modules

2021 ◽  
Vol 8 (29) ◽  
pp. 349-357
Author(s):  
David Benson ◽  
Jon Carlson
Keyword(s):  
Finite Group ◽  
Projective Modules ◽  
Content Type ◽  
One Dimensional ◽  
Direct Sum ◽  
Bounded Complex ◽  
Formula Group

Let k k be a field of characteristic p > 0 p > 0 . For G G an elementary abelian p p -group, there exist collections of permutation modules such that if C ∗ C^* is any exact bounded complex whose terms are sums of copies of modules from the collection, then C ∗ C^* is contractible. A consequence is that if G G is any finite group whose Sylow p p -subgroups are not cyclic or quaternion, and if C ∗ C^* is a bounded exact complex such that each C i C^i is a direct sum of one dimensional modules and projective modules, then C ∗ C^* is contractible.

Direct-sum decompositions over one-dimensional Cohen-Macaulay local rings

2006 ◽  
pp. 153-168 ◽  
Author(s):  
Alberto Facchini ◽  
Wolfgang Hassler ◽  
Lee Klingler ◽  
Roger Wiegand
Keyword(s):  
Local Rings ◽  
One Dimensional ◽  
Direct Sum

Some variations of projectivity

2021 ◽  
pp. 2250236
Author(s):  
Nil Orhan Ertaş ◽  
Rachid Tribak
Keyword(s):  
Middle Class ◽  
Integral Domain ◽  
Artinian Ring ◽  
Quotient Field ◽  
Commutative Noetherian Ring ◽  
One Dimensional ◽  
Direct Sum ◽  
Semisimple Modules ◽  
Injective Modules ◽  
Semisimple Ring

We prove that a ring [Formula: see text] has a module [Formula: see text] whose domain of projectivity consists of only some injective modules if and only if [Formula: see text] is a right noetherian right [Formula: see text]-ring. Also, we consider modules which are projective relative only to a subclass of max modules. Such modules are called max-poor modules. In a recent paper Holston et al. showed that every ring has a p-poor module (that is a module whose projectivity domain consists precisely of the semisimple modules). So every ring has a max-poor module. The structure of all max-poor abelian groups is completely determined. Examples of rings having a max-poor module which is neither projective nor p-poor are provided. We prove that the class of max-poor [Formula: see text]-modules is closed under direct summands if and only if [Formula: see text] is a right Bass ring. A ring [Formula: see text] is said to have no right max-p-middle class if every right [Formula: see text]-module is either projective or max-poor. It is shown that if a commutative noetherian ring [Formula: see text] has no right max-p-middle class, then [Formula: see text] is the ring direct sum of a semisimple ring [Formula: see text] and a ring [Formula: see text] which is either zero or an artinian ring or a one-dimensional local noetherian integral domain such that the quotient field [Formula: see text] of [Formula: see text] has a proper [Formula: see text]-submodule which is not complete in its [Formula: see text]-topology. Then we show that a commutative noetherian hereditary ring [Formula: see text] has no right max-p-middle class if and only if [Formula: see text] is a semisimple ring.

scholarly journals Commutative rings whose factors have Artinian rings of quotients

1983 ◽  
Vol 28 (1) ◽  
pp. 9-12 ◽  
Author(s):  
William D. Blair
Keyword(s):  
Commutative Ring ◽  
Quotient Ring ◽  
Commutative Rings ◽  
One Dimensional ◽  
Factor Ring ◽  
Artinian Rings ◽  
Direct Sum ◽  
Rings Of Quotients

Let R be a commutative ring with unity. Then every factor ring of R has an Artinian total quotient ring if and only if R is a direct sum of one-dimensional Noetherian domains and local Artinian rings.

0-Hecke algebras

1979 ◽  
Vol 27 (3) ◽  
pp. 337-357 ◽  
Author(s):  
P. N. Norton
Keyword(s):  
Coxeter Group ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Irreducible Representations ◽  
One Dimensional ◽  
Direct Sum ◽  
Natural Way

AbstractThe structure of a 0-Hecke algebra H of type (W, R) over a field is examined. H has 2n distinct irreducible representations, where n = ∣R∣, all of which are one-dimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2n indecomposable left ideals, in a similar way to Solomon's (1968) decomposition of the underlying Coxeter group W.

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