The behavior of ascending chain conditions on submodules of bounded finite generation in direct sums

2016 ◽  
Vol 215 (1) ◽  
pp. 339-347
Author(s):  
Pace P. Nielsen
Keyword(s):  
Finite Generation ◽  
Direct Sums ◽  
Chain Conditions

scholarly journals Equationally Compact Artinian Rings

1973 ◽  
Vol 25 (2) ◽  
pp. 273-283 ◽  
Author(s):  
David K. Haley
Keyword(s):  
Artinian Ring ◽  
Chain Condition ◽  
Finite Rings ◽  
Direct Sums ◽  
Artinian Rings ◽  
Chain Conditions ◽  
Descending Chain Condition ◽  
Equational Compactness ◽  
Theoretical Results

By a Noetherian (Artinian) ring=(R;+ , —, 0, ·) we mean an associative ring satisfying the ascending (descending) chain condition on left ideals. An arbitrary ringis said to beequationally compactif every system of ring polynomial equations with constants inis simultaneously solvable inprovided every finite subset is. (The reader is referred to [2; 8; 13; 14] for terminology and relevant results on equational compactness, and to [4] for unreferenced ring-theoretical results.) In this report a characterization of equationally compact Artinian rings is given - roughly speaking, these are the finite direct sums of finite rings and Prüfer groups; as consequences it is shown that an equationally compact ring satisfying both chain conditions is always finite, as is any Artinian ring which is a compact topological ring.

On Direct Sums of Injective Modules and Chain Conditions

1994 ◽  
Vol 46 (3) ◽  
pp. 634-647 ◽  
Author(s):  
Stanley S. Page ◽  
Yiqiang Zhou
Keyword(s):  
Full Subcategory ◽  
Natural Class ◽  
Direct Sums ◽  
Chain Conditions ◽  
Injective Modules ◽  
Injective Hulls ◽  
Equivalent Conditions

AbstractLet R be a ring and M a right R-module. Let σ[M] be the full subcategory of Mod-R subgenerated by M. An M-natural class 𝒦 is a subclass of σ[M] closed under submodules, direct sums, isomorphic copies, and M-injective hulls. We present some equivalent conditions each of which describes when σ has the property that direct sums of (M-)injective modules in σ are (M-)injective. Specializing to particular M, and/or special subclasses we obtain many new results and known results as corollaries.

scholarly journals Nonsingular bilinear forms on direct sums of ideals

2013 ◽  
Vol 63 (4) ◽  
Author(s):  
Beata Rothkegel
Keyword(s):  
Bilinear Form ◽  
Dedekind Domain ◽  
Bilinear Forms ◽  
Direct Sums ◽  
Direct Sum

AbstractIn the paper we formulate a criterion for the nonsingularity of a bilinear form on a direct sum of finitely many invertible ideals of a domain. We classify these forms up to isometry and, in the case of a Dedekind domain, up to similarity.

PERFECTLY BOUNDED CLASSES OF ABELIAN GROUPS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250208 ◽  
Author(s):  
PATRICK W. KEEF
Keyword(s):  
Abelian Groups ◽  
Direct Sums ◽  
Proper Subclass ◽  
Infinite Height ◽  
Reduced Group

Let [Formula: see text] be the class of abelian p-groups. A non-empty proper subclass [Formula: see text] is bounded if it is closed under subgroups, additively bounded if it is also closed under direct sums and perfectly bounded if it is additively bounded and closed under filtrations. If [Formula: see text], we call the partition of [Formula: see text] given by [Formula: see text] a B/U-pair. We state most of our results not in terms of bounded classes, but rather the corresponding B/U-pairs. Any additively bounded class contains a unique maximal perfectly bounded subclass. The idea of the length of a reduced group is generalized to the notion of the length of an additively bounded class. If λ is an ordinal or the symbol ∞, then there is a natural largest and smallest additively bounded class of length λ, as well as a largest and smallest perfectly bounded class of length λ. If λ ≤ ω, then there is a unique perfectly bounded class of length λ, namely the pλ-bounded groups that are direct sums of cyclics; however, this fails when λ > ω. This parallels results of Dugas, Fay and Shelah (1987) and Keef (1995) on the behavior of classes of abelian p-groups with elements of infinite height. It also simplifies, clarifies and generalizes a result of Cutler, Mader and Megibben (1989) which states that the pω + 1-projectives cannot be characterized using filtrations.

scholarly journals The Beckenbach-Dresher inequality in the Ψ-direct sums of spaces and related results

2012 ◽  
Vol 2012 (1) ◽  
pp. 7 ◽  
Author(s):  
Ludmila Nikolova ◽  
Lars-Erik Persson ◽  
Sanja Varošanec
Keyword(s):  
Direct Sums

scholarly journals N-th root rings

1987 ◽  
Vol 35 (1) ◽  
pp. 111-123 ◽  
Author(s):  
Henry Heatherly ◽  
Altha Blanchet
Keyword(s):  
Commutative Ring ◽  
Subdirect Product ◽  
General Construction ◽  
Fixed Integer ◽  
Chain Conditions

A ring for which there is a fixed integer n ≥ 2 such that every element in the ring has an n-th in the ring is called an n-th root ring. This paper gives numerous examples of diverse types of n-th root rings, some via general construction procedures. It is shown that every commutative ring can be embedded in a commutative n-th root ring with unity. The structure of n-th root rings with chain conditions is developed and finite n-th root rings are completely classified. Subdirect product representations are given for several classes of n-th root rings.

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