The Dilworth Number of Artinian Rings and Finite Posets with Rank Function

Abstract We introduce the notion of a rank function on a triangulated category 𝒞 {\mathcal{C}} which generalizes the Sylvester rank function in the case when 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} is the perfect derived category of a ring A. We show that rank functions are closely related to functors into simple triangulated categories and classify Verdier quotients into simple triangulated categories in terms of particular rank functions called localizing. If 𝒞 = 𝖯𝖾𝗋𝖿 ⁢ ( A ) {\mathcal{C}=\mathsf{Perf}(A)} as above, localizing rank functions also classify finite homological epimorphisms from A into differential graded skew-fields or, more generally, differential graded Artinian rings. To establish these results, we develop the theory of derived localization of differential graded algebras at thick subcategories of their perfect derived categories. This is a far-reaching generalization of Cohn’s matrix localization of rings and has independent interest.

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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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CONGRUENCES FOR ANDREWS' SMALLEST PARTS PARTITION FUNCTION AND NEW CONGRUENCES FOR DYSON'S RANK

International Journal of Number Theory ◽

10.1142/s179304211000296x ◽

2010 ◽

Vol 06 (02) ◽

pp. 281-309 ◽

Cited By ~ 32

Author(s):

F. G. GARVAN

Keyword(s):

Partition Function ◽

Rank Function ◽

Maass Forms ◽

Hecke Eigenforms ◽

Half Integer ◽

Integer Weight

Let spt (n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt (n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain explicit Ramanujan-type congruences for spt (n) mod ℓ for ℓ = 11, 17, 19, 29, 31 and 37. Recently, Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences. Their proof is non-constructive and utilizes the theory of weak Maass forms. We construct two explicit nontrivial examples mod 11 using elementary congruences between rank moments and half-integer weight Hecke eigenforms.

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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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