Completion of Modules over Serial, Right Noetherian Rings

2004 ◽  
Vol 120 (4) ◽  
pp. 1583-1590
Author(s):  
A. I. Generalov ◽  
I. M. Zilberbord
Keyword(s):  
Noetherian Rings ◽  
Right Noetherian Rings

Definability problems for modules and rings

1971 ◽  
Vol 36 (4) ◽  
pp. 623-649 ◽  
Author(s):  
Gabriel Sabbagh ◽  
Paul Eklof
Keyword(s):  
Ring Theory ◽  
Noetherian Rings ◽  
Fixed Ring ◽  
Injective Modules ◽  
Free Left ◽  
Right Noetherian Rings ◽  
The Right

This paper is concerned with questions of the following kind: let L be a language of the form Lαω and let be a class of modules over a fixed ring or a class of rings; is it possible to define in L? We will be mainly interested in the cases where L is Lωω or L∞ω and is a familiar class in homologic algebra or ring theory.In Part I we characterize the rings Λ such that the class of free (respectively projective, respectively flat) left Λ-modules is elementary. In [12] we solved the corresponding problems for injective modules; here we show that the class of injective Λ-modules is definable in L∞ω if and only if it is elementary. Moreover we identify the right noetherian rings Λ such that the class of projective (respectively free) left Λ-modules is definable in L∞ω.

3.—A Note on Projective Modules

1976 ◽  
Vol 75 (1) ◽  
pp. 24-31 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Noetherian Rings ◽  
Projective Modules ◽  
Finitely Generated ◽  
Right Noetherian Rings

SynopsisFor various classes of right noetherian rings it is shown that projective right modules are either finitely generated or free.

scholarly journals On maximal nilpotent subrings of right Noetherian rings

1967 ◽  
Vol 8 (2) ◽  
pp. 89-101 ◽  
Author(s):  
Gerhard Michler
Keyword(s):  
Artinian Ring ◽  
Noetherian Rings ◽  
Right Noetherian Rings ◽  
Nilpotent Subring ◽  
Unitary Right

Applying Hopkins's Theorem asserting that each unitary right Artinian ring is right Noetherian, G. Köthe and K. Shoda proved the following theorem (cf. Köthe [7], p. 360, Theorem 1 and p. 363, Theorem 5): If R is a unitary right Artinian ring, then the following statements hold:(i) Each nilpotent subring of R is contained in a maximal nilpotent subring of R.(ii) The intersection of all maximal nilpotent subrings of R is the maximal nilpotent twosided ideal of R.(iii) All maximal nilpotent subrings of R are conjugate.

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