Endomorphism rings of essential extensions of a noetherian module

1974 ◽  
Vol 17 (1) ◽  
pp. 46-49 ◽  
Author(s):  
M. G. Deshpande ◽  
E. H. Feller
1980 ◽  
Vol 32 (6) ◽  
pp. 1277-1287 ◽  
Author(s):  
B. Sarath

The main object of this paper is to study when infective noetherian modules are artinian. This question was first raised by J. Fisher and an example of an injective noetherian module which is not artinian is given in [9]. However, it is shown in [4] that over commutative rings, and over hereditary noetherian P.I rings, injective noetherian does imply artinian. By combining results of [6] and [4] it can be shown that the above implication is true over any noetherian P.I ring. It is shown in this paper that injective noetherian modules are artinian over rings finitely generated as modules over their centers, and over semiprime rings of Krull dimension 1. It is also shown that every injective noetherian module over a P.I ring contains a simple submodule. Since any noetherian injective module is a finite direct sum of indecomposable injectives it suffices to study when such injectives are artinian. IfQis an injective indecomposable noetherian module, thenQcontains a non-zero submoduleQ0such that the endomorphism rings ofQ0and all its submodules are skewfields. Over a commutative ring, such aQ0is simple. In the non-commutative case very little can be concluded, and many of the difficulties seem to arise here.


2021 ◽  
Vol 18 (4) ◽  
Author(s):  
Manuel Cortés-Izurdiaga ◽  
Pedro A. Guil Asensio ◽  
D. Keskin Tütüncü ◽  
Ashish K. Srivastava
Keyword(s):  

Author(s):  
Claus Fieker ◽  
Tommy Hofmann ◽  
Sogo Pierre Sanon

1994 ◽  
Vol 121 (2) ◽  
pp. 335-335 ◽  
Author(s):  
A. W. Chatters ◽  
C. R. Hajarnavis

2002 ◽  
Vol 67 (2) ◽  
pp. 635-648
Author(s):  
Xavier Vidaux

AbstractLet K and K′ be two elliptic fields with complex multiplication over an algebraically closed field k of characteristic 0. non k-isomorphic, and let C and C′ be two curves with respectively K and K′ as function fields. We prove that if the endomorphism rings of the curves are not isomorphic then K and K′ are not elementarily equivalent in the language of fields expanded with a constant symbol (the modular invariant). This theorem is an analogue of a theorem from David A. Pierce in the language of k-algebras.


1987 ◽  
Vol 35 (3) ◽  
pp. 379-386 ◽  
Author(s):  
E. R. Puczyłwski

This paper concerns the problem of description of the set of rings containing a given ring as an essential ideal. The results obtained are applied to some problems of ring theory and radicals.


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