cyclic module
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2017 ◽  
Vol 16 (02) ◽  
pp. 1750024
Author(s):  
C. Selvaraj ◽  
S. Santhakumar

In this paper, we investigate some properties of dual automorphism invariant modules over right perfect rings. Also, we introduce the notion of dual automorphism invariant cover and prove the existence of dual automorphism invariant cover. Moreover, we give the necessary and sufficient condition for every cyclic module to be a dual automorphism invariant module over a semi perfect ring and we prove that supplemented quasi projective module has finite exchange property. Also we give a characterization of a perfect ring using dual automorphism invariant module.


2016 ◽  
Vol 59 (3) ◽  
pp. 624-640
Author(s):  
Noriyuki Otsubo

AbstractWe give a precise description of the homology group of the Fermat curve as a cyclic module over a group ring. As an application, we prove the freeness of the profinite homology of the Fermat tower. This allows us to define measures, an equivalent of Anderson’s adelic beta functions, in a manner similar to Ihara’s definition of ℓ-adic universal power series for Jacobi sums. We give a simple proof of the interpolation property using a motivic decomposition of the Fermat curve.


2016 ◽  
Vol 15 (05) ◽  
pp. 1650078 ◽  
Author(s):  
M. Tamer Koşan ◽  
Nguyen Thi Thu Ha ◽  
Truong Cong Quynh

Rings all of whose right ideals are automorphism-invariant are called right [Formula: see text]-rings. In the present paper, we study rings having the property that every right cyclic module is dual automorphism-invariant. Such rings are called right [Formula: see text]-rings. We obtain some of the relationships [Formula: see text]-rings and [Formula: see text]-rings. We also prove that; (i) A semiperfect ring [Formula: see text] is a right [Formula: see text]-ring if and only if any right ideal in [Formula: see text] is a left [Formula: see text]-module, where [Formula: see text] is a subring of [Formula: see text] generated by its units, (ii) [Formula: see text] is semisimple artinian if and only if [Formula: see text] is semiperfect and the matrix ring [Formula: see text] is a right [Formula: see text]-ring for all [Formula: see text], (iii) Quasi-Frobenius right [Formula: see text]-rings are Frobenius.


2014 ◽  
Vol 14 (02) ◽  
pp. 1550021
Author(s):  
Sebastian Burciu

It is shown that any coideal subalgebra of a finite-dimensional Hopf algebra is a cyclic module over the dual Hopf algebra. Using this we describe all coideal subalgebras of a cocentral abelian extension of Hopf algebras extending some results from [R. Guralnick and F. Xu, On a subfactor generalization of Wall's conjecture, J. Algebra 332 (2011) 457–468].


2012 ◽  
Vol 12 (01) ◽  
pp. 1250127 ◽  
Author(s):  
SARAPEE CHAIRAT ◽  
DINH VAN HUYNH ◽  
CHITLADA SOMSUP

Carl Faith (2003) introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings). Inspired by this we investigate rings over which every cyclic right R-module has a Σ-extending injective hull. We call such rings right CSE-rings and show that the class of right CSE-rings and that of right CSI-rings coincide. We also use other hulls of cyclic modules to define other classes of rings, and investigate their structure. We prove, among others, that a ring R is right QI if and only if the quasi-injective hull of each cyclic right module is Σ-injective.


2012 ◽  
Vol 54 (3) ◽  
pp. 605-617 ◽  
Author(s):  
PINAR AYDOĞDU ◽  
NOYAN ER ◽  
NİL ORHAN ERTAŞ

AbstractDedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.


2010 ◽  
Vol 52 (A) ◽  
pp. 61-67 ◽  
Author(s):  
SEPTIMIU CRIVEI ◽  
CONSTANTIN NĂSTĂSESCU ◽  
BLAS TORRECILLAS

AbstractWe recall a version of the Osofsky–Smith theorem in the context of a Grothendieck category and derive several consequences of this result. For example, it is deduced that every locally finitely generated Grothendieck category with a family of completely injective finitely generated generators is semi-simple. We also discuss the torsion-theoretic version of the classical Osofsky theorem which characterizes semi-simple rings as those rings whose every cyclic module is injective.


K-Theory ◽  
2002 ◽  
Vol 27 (2) ◽  
pp. 111-131 ◽  
Author(s):  
M. Khalkhali ◽  
B. Rangipour
Keyword(s):  

1995 ◽  
Vol 52 (3) ◽  
pp. 517-525 ◽  
Author(s):  
Yiqiang Zhou

Responding to a question on right weakly semisimple rings due to Jain, Lopez-Permouth and Singh, we report the existence of a non-right-Noetherian ring R for which every uniform cyclic right it-module is weakly-injective and every uniform finitely generated right R-module is compressible. We show that a ring R is a right Noetherian ring for which every cyclic right R-module is weakly R-injective if and only if R is a right Noetherian ring for which every uniform cyclic right R-module is compressible if and only if every cyclic right R-module is compressible. Finally, we characterise those modules M for which every finitely generated (respectively, cyclic) module in σ[M] is compressible.


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