scholarly journals Different Characterizations of Large Submodules of QTAG-Modules

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Fahad Sikander ◽  
Alveera Mehdi ◽  
Sabah A. R. K. Naji
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Necessary And Sufficient

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. The study of large submodules and its fascinating properties makes the theory of QTAG-modules more interesting. A fully invariant submodule L of M is large in M if L+B=M, for every basic submodule B of M. The impetus of these efforts lies in the fact that the rings are almost restriction-free. This motivates us to find the necessary and sufficient conditions for a submodule of a QTAG-module to be large and characterize them. Also, we investigate some properties of large submodules shared by Σ-modules, summable modules, σ-summable modules, and so on.

scholarly journals Nice Bases of QTAG-Modules

2021 ◽  
pp. 1-5
Author(s):  
Fahad Sikander ◽  
Tanveer Fatima ◽  
Ayazul Hasan
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Divisible Hull

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of universal modules. In this paper, we investigate the class of QTAG-modules having nice basis. It is proved that if H_ω (M) is bounded then M has a bounded nice basis and if H_ω (M) is a direct sum of uniserial modules, then M has a nice basis. We also proved that if M is any QTAG-module, then M⊕D has a nice basis, where D is the h-divisible hull of H_ω (M).

scholarly journals On Quasi-h-Dense Submodules and h-Pure Envelopes of QTAG Modules

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Alveera Mehdi ◽  
Fahad Sikander ◽  
Firdhousi Begum
Keyword(s):  
Direct Summand ◽  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules ◽  
Pure Submodules

A module M over an associative ring R with unity is a QTAG module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. There are many fascinating properties of QTAG modules of which h-pure submodules and high submodules are significant. A submodule N is quasi-h-dense in M if M/K is h-divisible, for every h-pure submodule K of M, containing N. Here we study these submodules and obtain some interesting results. Motivated by h-neat envelope, we also define h-pure envelope of a submodule N as the h-pure submodule K⊇N if K has no direct summand containing N. We find that h-pure envelopes of N have isomorphic basic submodules, and if M is the direct sum of uniserial modules, then all h-pure envelopes of N are isomorphic.

Structures of QTAG-modules determined by their submodules

2021 ◽  
Vol 65 (3) ◽  
pp. 38-45
Author(s):  
Ayazul Hasan
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules

A module M over an associative ring R with unity is a QTAG-module if every finitely generated submodule of any homomorphic image of M is a direct sum of uniserial modules. In this paper, we study the existence of several classes C of QTAG-modules which satisfy the property that M belongs to C uniquely when M/N belongs to C provided that N is a finitely generated submodule of the QTAG-module.

scholarly journals A Note on Elongations of SummableQTAG-Modules

2013 ◽  
Vol 2013 ◽  
pp. 1-3 ◽  
Author(s):  
Alveera Mehdi ◽  
Fahad Sikander ◽  
Sabah A. R. K. Naji
Keyword(s):  
Homomorphic Image ◽  
Associative Ring ◽  
Suitable Condition ◽  
Finitely Generated ◽  
Direct Sum ◽  
Uniserial Modules

A right moduleMover an associative ring with unity is aQTAG-module if every finitely generated submodule of any homomorphic image ofMis a direct sum of uniserial modules. In this paper we find a suitable condition under which a specialω-elongation of a summableQTAG-module by aω+k-projectiveQTAG-module is also a summableQTAG-module.

scholarly journals Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Brendan Goldsmith ◽  
Ketao Gong
Keyword(s):  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sums ◽  
Direct Sum ◽  
Zero Entropy ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.

scholarly journals Inertial subalgebras of algebras possessing finite automorphism groups

1979 ◽  
Vol 28 (3) ◽  
pp. 335-345 ◽  
Author(s):  
Nicholas S. Ford
Keyword(s):  
Finite Group ◽  
Commutative Ring ◽  
Jacobson Radical ◽  
Automorphism Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Fixed Ring ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet R be a commutative ring with identity, and let A be a finitely generated R-algebra with Jacobson radical N and center C. An R-inertial subalgebra of A is a R-separable subalgebra B with the property that B+N=A. Suppose A is separable over C and possesses a finite group G of R-automorphisms whose restriction to C is faithful with fixed ring R. If R is an inertial subalgebra of C, necessary and sufficient conditions for the existence of an R-inertial subalgebra of A are found when the order of G is a unit in R. Under these conditions, an R-inertial subalgebra B of A is characterized as being the fixed subring of a group of R-automorphisms of A. Moreover, A ⋍ B ⊗R C. Analogous results are obtained when C has an R-inertial subalgebra S ⊃ R.

scholarly journals On the equivalence of cancellative extensions of commutative cancellative semigroups by groups

1971 ◽  
Vol 12 (2) ◽  
pp. 187-192
Author(s):  
Charles V. Heuer
Keyword(s):  
Abelian Group ◽  
Cyclic Group ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ideal Extension ◽  
Cancellative Semigroup ◽  
Finitely Generated ◽  
Necessary And Sufficient

In [1] D. W. Miller and the author established necessary and sufficient conditions for the existence of a cancellative (ideal) extension of a commutative cancellative semigroup by a cyclic group with zero. The purpose of this paper is to extend these results to cancellative extensions by any finitely generated Abelian group with zero and to establish in this general case conditions under which two such extensions are equivalent.

Isomorphisms of Prime Goldie Semi-Principal Left Ideal Rings, II

1988 ◽  
Vol 31 (3) ◽  
pp. 374-379 ◽  
Author(s):  
Kenneth G. Wolfson
Keyword(s):  
Left Ideal ◽  
Endomorphism Ring ◽  
Finite Rank ◽  
Sufficient Conditions ◽  
Free Module ◽  
Necessary And Sufficient Conditions ◽  
Finitely Generated ◽  
Ore Domain ◽  
Necessary And Sufficient

AbstractA prime Goldie ring K, in which each finitely generated left ideal is principal is the endomorphism ring E(F, A) of a free module A, of finite rank, over an Ore domain F. We determine necessary and sufficient conditions to insure that whenever K ≅ E(F, A) ≅ E(G, B) (with A and B free and finitely generated over domains F and G) then (F, A) is semi-linearly isomorphic to (G, B). We also show, by example, that it is possible for K ≅ E(F, A ) ≅ E(G, B), with F and G, not isomorphic.

Hilbert rings and G(oldman)-rings issued from amalgamated algebras

2018 ◽  
Vol 17 (02) ◽  
pp. 1850023 ◽  
Author(s):  
L. Izelgue ◽  
O. Ouzzaouit
Keyword(s):  
New York ◽  
Commutative Algebra ◽  
Quotient Ring ◽  
Sufficient Conditions ◽  
Ring Homomorphism ◽  
Necessary And Sufficient Conditions ◽  
Local Rings ◽  
Finitely Generated ◽  
Necessary And Sufficient ◽  
The Way

Let [Formula: see text] and [Formula: see text] be two rings, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] be a ring homomorphism. The ring [Formula: see text] is called the amalgamation of [Formula: see text] with [Formula: see text] along [Formula: see text] with respect to [Formula: see text]. It was proposed by D’anna and Fontana [Amalgamated algebras along an ideal, Commutative Algebra and Applications (W. de Gruyter Publisher, Berlin, 2009), pp. 155–172], as an extension for the Nagata’s idealization, which was originally introduced in [Nagata, Local Rings (Interscience, New York, 1962)]. In this paper, we establish necessary and sufficient conditions under which [Formula: see text], and some related constructions, is either a Hilbert ring, a [Formula: see text]-domain or a [Formula: see text]-ring in the sense of Adams [Rings with a finitely generated total quotient ring, Canad. Math. Bull. 17(1) (1974)]. By the way, we investigate the transfer of the [Formula: see text]-property among pairs of domains sharing an ideal. Our results provide original illustrating examples.

NORMALITY OF f-UNITARY UNITS IN AN ALTERNATIVE LOOP RING

2006 ◽  
Vol 05 (04) ◽  
pp. 537-548
Author(s):  
EDGAR G. GOODAIRE ◽  
CÉSAR POLCINO MILIES
Keyword(s):  
Associative Ring ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Loop Ring ◽  
Necessary And Sufficient

Let L be an RA loop, that is, a loop whose loop ring in any characteristic is an alternative, but not associative, ring. Let f : L → {±1} be a homomorphism and for α = ∑αℓℓ in the integral loop ring Z L, define αf = ∑αℓf(ℓ)ℓ-1. A unit u ∈ Z L is said to be f-unitary if uf = ±u-1. The set [Formula: see text] of all f-unitary units is a subloop of [Formula: see text], the loop of all units in Z L. In this paper, we find necessary and sufficient conditions for [Formula: see text] to be normal in [Formula: see text].

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