Noetherian and Artinian Rings. Primary Decomposition

AbstractLet S = K[x1 , . . . , xn] be the polynomial ring in n-variables over a ûeld K and I a monomial ideal of S. According to one standard primary decomposition of I, we get a Stanley decomposition of the monomial factor algebra S/I. Using this Stanley decomposition, one can estimate the Stanley depth of S/I. It is proved that sdepthS(S/I) ≤ sizeS(I). When I is squarefree and bigsizeS(I) ≤ 2, the Stanley conjecture holds for S/I, i.e., sdepthS(S/I) ≥ depthS(S/I).

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Artinian bimodule with quasi-Frobenius bimodule of translations

Discrete Mathematics and Applications ◽

10.1515/dma-2019-0010 ◽

2019 ◽

Vol 29 (2) ◽

pp. 103-119

Author(s):

Aleksandr A. Nechaev ◽

Vadim N. Tsypyschev

Keyword(s):

Commutative Ring ◽

Recurrent Sequence ◽

Artinian Rings ◽

Linear Recurrent Sequence ◽

Left And Right ◽

Additional Assumptions

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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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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