scholarly journals α-modules and generalized submodules

2019 ◽  
Vol 27 (1) ◽  
pp. 13-26
Author(s):  
◽  
Ayazul Hasan ◽  
Mohammad Fareed Ahmad
Keyword(s):  
Limit Ordinal ◽  
Pure Submodules

AbstractA QTAG-module M is an α-module, where α is a limit ordinal, if M/Hβ (M) is totally projective for every ordinal β < α. In the present paper α-modules are studied with the help of α-pure submodules, α-basic submodules, and α-large submodules. It is found that an α-closed α-module is an α-injective. For any ordinal ω ≤ α ≤ ω1 we prove that an α-large submodule L of an ω1-module M is summable if and only if M is summable.

α-degrees of α-theories

1972 ◽  
Vol 37 (4) ◽  
pp. 677-682 ◽  
Author(s):  
George Metakides
Keyword(s):  
Initial Segment ◽  
Recursion Theory ◽  
The Other ◽  
Infinite Length ◽  
Infinitary Logic ◽  
Admissible Set ◽  
Recursively Enumerable ◽  
Limit Ordinal ◽  
The Subject ◽  
Further Development

Let α be a limit ordinal with the property that any “recursive” function whose domain is a proper initial segment of α has its range bounded by α. α is then called admissible (in a sense to be made precise later) and a recursion theory can be developed on it (α-recursion theory) by providing the generalized notions of α-recursively enumerable, α-recursive and α-finite. Takeuti [12] was the first to study recursive functions of ordinals, the subject owing its further development to Kripke [7], Platek [8], Kreisel [6], and Sacks [9].Infinitary logic on the other hand (i.e., the study of languages which allow expressions of infinite length) was quite extensively studied by Scott [11], Tarski, Kreisel, Karp [5] and others. Kreisel suggested in the late '50's that these languages (even which allows countable expressions but only finite quantification) were too large and that one should only allow expressions which are, in some generalized sense, finite. This made the application of generalized recursion theory to the logic of infinitary languages appear natural. In 1967 Barwise [1] was the first to present a complete formalization of the restriction of to an admissible fragment (A a countable admissible set) and to prove that completeness and compactness hold for it. [2] is an excellent reference for a detailed exposition of admissible languages.

Strict Mittag–Leffler modules over Gorenstein injective modules

2019 ◽  
Vol 19 (03) ◽  
pp. 2050050 ◽  
Author(s):  
Yanjiong Yang ◽  
Xiaoguang Yan
Keyword(s):  
Noetherian Ring ◽  
Direct Limit ◽  
Injective Module ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Injective Modules ◽  
Pure Submodules ◽  
Covering Class ◽  
Gorenstein Injective ◽  
Finitely Presented

In this paper, we study the conditions under which a module is a strict Mittag–Leffler module over the class [Formula: see text] of Gorenstein injective modules. To this aim, we introduce the notion of [Formula: see text]-projective modules and prove that over noetherian rings, if a module can be expressed as the direct limit of finitely presented [Formula: see text]-projective modules, then it is a strict Mittag–Leffler module over [Formula: see text]. As applications, we prove that if [Formula: see text] is a two-sided noetherian ring, then [Formula: see text] is a covering class closed under pure submodules if and only if every injective module is strict Mittag–Leffler over [Formula: see text].

scholarly journals A Generalization of Final Rank of Primary Abelian Groups

1970 ◽  
Vol 22 (6) ◽  
pp. 1118-1122 ◽  
Author(s):  
Doyle O. Cutler ◽  
Paul F. Dubois
Keyword(s):  
Abelian Group ◽  
Abelian Groups ◽  
Basic Subgroup ◽  
Primary Abelian Group ◽  
Limit Ordinal ◽  
Final Rank ◽  
Definition Of ◽  
Group Recall

Let G be a p-primary Abelian group. Recall that the final rank of G is infn∈ω{r(pnG)}, where r(pnG) is the rank of pnG and ω is the first limit ordinal. Alternately, if Γ is the set of all basic subgroups of G, we may define the final rank of G by supB∈Γ {r(G/B)}. In fact, it is known that there exists a basic subgroup B of G such that r(G/B) is equal to the final rank of G. Since the final rank of G is equal to the final rank of a high subgroup of G plus the rank of pωG, one could obtain the same information if the definition of final rank were restricted to the class of p-primary Abelian groups of length ω.

A Theorem on Pure Submodules

1960 ◽  
Vol 12 ◽  
pp. 483-487
Author(s):  
George Kolettis
Keyword(s):  
Free Abelian Group ◽  
Chain Condition ◽  
Affirmative Answer ◽  
Principal Ideal Ring ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Rank One ◽  
Pure Submodules ◽  
Torsion Free ◽  
Countable Case

In (1) Baer studied the following problem: If a torsion-free abelian group G is a direct sum of groups of rank one, is every direct summand of G also a direct sum of groups of rank one? For groups satisfying a certain chain condition, Baer gave a solution. Kulikov, in (3), supplied an affirmative answer, assuming only that G is countable. In a recent paper (2), Kaplansky settles the issue by reducing the general case to the countable case where Kulikov's solution is applicable. As usual, the result extends to modules over a principal ideal ring R (commutative with unit, no divisors of zero, every ideal principal).The object of this paper is to carry out a similar investigation for pure submodules, a somewhat larger class of submodules than the class of direct summands. We ask: if the torsion-free i?-module M is a direct sum of modules of rank one, is every pure submodule N of M also a direct sum of modules of rank one? Unlike the situation for direct summands, here the answer depends heavily on the ring R.

scholarly journals On Crossed Products of a Sfield

1963 ◽  
Vol 22 ◽  
pp. 65-71 ◽  
Author(s):  
Masatoshi Ikeda
Keyword(s):  
Integral Domain ◽  
Free Abelian Group ◽  
Group Extension ◽  
Crossed Products ◽  
Torsion Free Abelian Group ◽  
Outer Automorphisms ◽  
Identity Automorphism ◽  
Limit Ordinal ◽  
Group B ◽  
Torsion Free

In the previous paper [3] the author has shown a possibility to construct a series of sfields by taking sfields of quotients of split crossed products of a sfield. In this paper the same problem is treated, and, by considering general crossed products, a generalization of the previous result is given: Let K be a sfield and G be the join of a well-ordered ascending chain of groups Gα of outer automorphisms of K such that a) G1 is the identity automorphism group, b) Gα is a group extension of Gα-1 by a torsion-free abelian group for each non-limit ordinal α, and c) for each limit ordinal α. Then an arbitrary crossed product of K with G is an integral domain with a sfield of quotients Q and the commutor ring of K in Q coincides with the centre of K.

Intrinsic bounds on complexity and definability at limit levels

2009 ◽  
Vol 74 (3) ◽  
pp. 1047-1060 ◽  
Author(s):  
John Chisholm ◽  
Ekaterina B. Fokina ◽  
Sergey S. Goncharov ◽  
Valentina S. Harizanov ◽  
Julia F. Knight ◽  
...  
Keyword(s):  
Computable Structure ◽  
Additional Relation ◽  
Scott Family ◽  
Limit Ordinal

AbstractWe show that for every computable limit ordinal α, there is a computable structure that is categorical, but not relatively categorical (equivalently, it does not have a formally Scott family). We also show that for every computable limit ordinal α, there is a computable structure with an additional relation R that is intrinsically on , but not relatively intrinsically on (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.

scholarly journals Pure submodules

1967 ◽  
Vol 7 (2) ◽  
pp. 159-171 ◽  
Author(s):  
Bo T. Stenström
Keyword(s):  
Pure Submodules

Pure submodules of modules over bounded (hnp)-rings

1978 ◽  
Vol 30 (1) ◽  
pp. 570-577 ◽  
Author(s):  
Surjeet Singh ◽  
Sudha Talwar
Keyword(s):  
Pure Submodules

scholarly journals s-pure submodules

2005 ◽  
Vol 2005 (4) ◽  
pp. 491-497 ◽  
Author(s):  
Iuliu Crivei
Keyword(s):  
Maximal Ideal ◽  
Commutative Rings ◽  
Pure Submodules

A submoduleAof a rightR-moduleBis calleds-pure iff⊗R1Sis a monomorphism for every simple leftR-moduleS, wheref:A→Bis the inclusion homomorphism. We establish some properties ofs-pure submodules and uses-purity to characterize commutative rings with every maximal ideal idempotent.

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