Representations of Quivers over Artinian Rings

Abstract The possibility to generalize the notion of a linear recurrent sequence (LRS) over a commutative ring to the case of a LRS over a non-commutative ring is discussed. In this context, an arbitrary bimodule AMB over left- and right-Artinian rings A and B, respectively, is associated with the equivalent bimodule of translations CMZ, where C is the multiplicative ring of the bimodule AMB and Z is its center, and the relation between the quasi-Frobenius conditions for the bimodules AMB and CMZ is studied. It is demonstrated that, in the general case, the fact that AMB is a quasi-Frobenius bimodule does not imply the validity of the quasi-Frobenius condition for the bimodule CMZ. However, under some additional assumptions it can be shown that if CMZ is a quasi-Frobenius bimodule, then the bimodule AMB is quasi-Frobenius as well.

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Strongly Irreducible Operators and Indecomposable Representations of Quivers on Infinite-Dimensional Hilbert Spaces

Integral Equations and Operator Theory ◽

10.1007/s00020-015-2228-3 ◽

2015 ◽

Vol 83 (4) ◽

pp. 563-587

Author(s):

Masatoshi Enomoto ◽

Yasuo Watatani

Keyword(s):

Hilbert Spaces ◽

Infinite Dimensional ◽

Representations Of Quivers ◽

Indecomposable Representations ◽

Strongly Irreducible Operators ◽

Strongly Irreducible

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Projective representations of quivers

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171202108192 ◽

2002 ◽

Vol 31 (2) ◽

pp. 97-101 ◽

Cited By ~ 1

Author(s):

Sangwon Park

Keyword(s):

Projective Representation ◽

Representations Of Quivers ◽

Direct Sum ◽

Projective Representations

We prove thatP1 →f P2is a projective representation of a quiverQ=•→•if and only ifP1andP2are projective leftR-modules,fis an injection, andf (P 1)⊂P 2is a summand. Then, we generalize the result so that a representationM1 →f1 M2 →f2⋯→fn−2 Mn−1→fn−1 Mnof a quiverQ=•→•→•⋯•→•→•is projective representation if and only if eachMiis a projective leftR-module and the representation is a direct sum of projective representations.

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A GENERALIZATION OF SEMISIMPLE ARTINIAN RINGS

Journal of Algebra and Its Applications ◽

10.1142/s0219498805001137 ◽

2005 ◽

Vol 04 (03) ◽

pp. 231-235

Author(s):

YASUYUKI HIRANO ◽

HISAYA TSUTSUI

Keyword(s):

Direct Summand ◽

Artinian Rings

We investigate a ring R with the property that for every right R-module M and every ideal I of R the annihilator of I in M is a direct summand of M, and determine conditions under which such a ring is semisimple Artinian.

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On commutative rings with uniserial dimension

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500085 ◽

2014 ◽

Vol 14 (01) ◽

pp. 1550008 ◽

Cited By ~ 2

Author(s):

A. Ghorbani ◽

Z. Nazemian

Keyword(s):

Commutative Ring ◽

Commutative Rings ◽

Artinian Rings ◽

Uniform Dimension ◽

Uniserial Modules

In this paper, we define and study a valuation dimension for commutative rings. The valuation dimension is a measure of how far a commutative ring deviates from being valuation. It is shown that a ring R with valuation dimension has finite uniform dimension. We prove that a ring R is Noetherian (respectively, Artinian) if and only if the ring R × R has (respectively, finite) valuation dimension if and only if R has (respectively, finite) valuation dimension and all cyclic uniserial modules are Noetherian (respectively, Artinian). We show that the class of all rings of finite valuation dimension strictly lies between the class of Artinian rings and the class of semi-perfect rings.

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