Some variants of ascending and descending chain conditions

2021 ◽  
pp. 1-10
Author(s):  
Avanish Kumar Chaturvedi ◽  
Surya Prakash
Keyword(s):  
Chain Conditions

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.

Chain Conditions

2004 ◽  
pp. 173-176
Author(s):  
Vesselin Drensky ◽  
Edward Formanek
Keyword(s):  
Chain Conditions

scholarly journals Rings whose additive endomorphisms are ring endomorphisms

1990 ◽  
Vol 42 (1) ◽  
pp. 145-152 ◽  
Author(s):  
Gary Birkenmeier ◽  
Henry Heatherly
Keyword(s):  
Subdirectly Irreducible ◽  
Finiteness Conditions ◽  
Chain Conditions ◽  
Open Questions ◽  
Open Case ◽  
Ring Endomorphism ◽  
Substantial Progress ◽  
Additive Endomorphism ◽  
Made In

A ring R is said to be an AE-ring if every additive endomorphism is a ring endomorphism. In this paper further steps are made toward solving Sullivan's Problem of characterising these rings. The classification of AE-rings with. R3 ≠ 0 is completed. Complete characterisations are given for AE-rings which are either: (i) subdirectly irreducible, (ii) algebras over fields, or (iii) additively indecomposable. Substantial progress is made in classifying AE-rings which are mixed – the last open case – by imposing various finiteness conditions (chain conditions on special ideals, height restricting conditions). Several open questions are posed.

CHAIN CONDITIONS ON NON-SUMMANDS

2011 ◽  
Vol 10 (03) ◽  
pp. 475-489 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
A. ÇIĞDEM ÖZCAN ◽  
PATRICK F. SMITH
Keyword(s):  
Noetherian Ring ◽  
Jacobson Radical ◽  
Finitely Generated ◽  
Chain Conditions ◽  
Finite Dimensional

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes [Formula: see text], such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

Descending Chain Conditions

2020 ◽  
pp. 159-178
Author(s):  
Jan Okniński
Keyword(s):  
Chain Conditions
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