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Model theory of modules over a serial ring

Paul C. Eklof, Ivo Herzog
1995 Annals of Pure and Applied Logic  
We use the Drozd-Warfield structure theorem for finitely presented modules over a serial ring to investigate the model theory of modules over a serial ring, in particular, to give a simple description  ...  We also study the question of whether every pure-injective indecomposable module over a valuation ring is the hull of a uniserial module. 0168~0072/95/%09.50 0 1995 -Elsevier Science B.V.  ...  We claim that the image, a, of 1 + I E R/Z = R/Ir, in U realizes the type (I, J) in U. Because the maps in the direct limit are one-one, ann (a) = I.  ... 
doi:10.1016/0168-0072(94)00017-w fatcat:akp5vvrukvcenmln7t5otdu2dm

Page 6012 of Mathematical Reviews Vol. , Issue 99i [page]

1999 Mathematical Reviews  
So does module theory over serial rings, which is one of the meeting points of algebra and model theory. It is therefore a pity that the authors have not published their book in English.  ...  each finitely presented module over a serial ring is serial.  ... 

Page 906 of Mathematical Reviews Vol. , Issue 2003B [page]

2003 Mathematical Reviews  
Some model theory over an exceptional uniserial ring and decompositions of serial modules. (English summary) J. London Math. Soc. (2) 64 (2001), no. 2, 311-326.  ...  of a serial module.  ... 

Page 2588 of Mathematical Reviews Vol. , Issue 96e [page]

1996 Mathematical Reviews  
The authors use the Drozd-Warfield structure theorem for finitely presented modules over serial rings to investigate the model the- ory of modules over such rings.  ...  Mike Prest (4-MANC; Manchester) 96e:03041 03C60 13F30 16D5S0 Eklof, Paul C. (1-CA3; Irvine, CA); Herzog, Ivo (1-BRND; Waltham, MA) Model theory of modules over a serial ring. (English summary) Ann.  ... 

The Krull–Gabriel Dimension of a Serial Ring

Gena Puninski
2003 Communications in Algebra  
We prove that every serial ring R has the isolation property: every isolated point in any theory of modules over R is isolated by a minimal pair.  ...  Using this we calculate the Krull-Gabriel dimension of the module category over serial rings.  ...  Corollary 3. 5 . 5 The isolation property holds for every theory of modules over a commutative Prüfer ring. Proposition 3. 6 . 6 Let R be a serial ring.  ... 
doi:10.1081/agb-120024862 fatcat:txvhtirgmvgg5oq5voktgqhk4q

Page 7123 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
The second half of the book investigates the structure theory of modules over a serial ring, including: model theory for modules, indecomposable pure injective modules over a serial ring, super decomposable  ...  pure injective modules over a commutative valuation ring (or domain), pure projective modules over certain uniserial rings, and finally, i-pure injective modules over serial rings.  ... 

Page 5247 of Mathematical Reviews Vol. , Issue 96i [page]

1996 Mathematical Reviews  
A complete description of pure-injective modules over a Noetherian serial ring was obtained by A. I. Generalov [in Rings and modules.  ...  This interesting paper combines well-known results on these classes of rings with results proved using model theory for modules to prove the following result: The following are equivalent for a serial  ... 

Page 6066 of Mathematical Reviews Vol. , Issue 2001I [page]

2001 Mathematical Reviews  
The proof uses the model theory of modules over serial rings as developed by P. C. Eklof and I. Herzog [Ann. Pure Appl.  ...  Let R be a serial ring (that is, R decomposes, as a right and as a left module, as a direct sum of modules each of which has a linearly ordered lattice of submodules) which is countable.  ... 

Vaught's Conjecture for Modules Over a Dedekind Prime Ring

Vera Puninskaya
1999 Bulletin of the London Mathematical Society  
Ziegler [15] checked Vaught's conjecture for complete theories of modules over a countable commutative Dedekind domain, and Puninskaya [12] verified the similar result for modules over a countable serial  ...  Baldwin and McKenzie proved that Vaught's conjecture is true for the theory of all modules over a countable ring [1] .  ...  A module that is a direct sum of uniserial modules is called serial. We say that a ring R is serial if R considered as a right R R and a left R R module over itself is serial. E 1.1.  ... 
doi:10.1112/s0024609398005360 fatcat:7p4jqxlv2fc4vjb3vxnm5y2hbu

Page 2318 of Mathematical Reviews Vol. , Issue 2003d [page]

2003 Mathematical Reviews  
Recall that a module M over a countable ring is £-pure-injective if and only if the complete theory T = Th(M) is w-stable.  ...  The main body of the paper is devoted to a largely self-contained survey of what is presently known for the case when T is a complete theory of modules over a countable ring R.  ... 

Page 2296 of Mathematical Reviews Vol. , Issue 2001D [page]

2001 Mathematical Reviews  
A serial module is defined as a direct sum of modules, each of which has the property that its lattice of submodules is a chain.  ...  The author begins by giving interpretations and proofs of var- ious of Crawley-Boevey’s results on characters in terms of the model theory of modules.  ... 

Some Model Theory Over an Exceptional Uniserial Ring and Decompositions of Serial Modules

Gennadi Puninski
2001 Journal of the London Mathematical Society  
We give an example of a direct summand of a serial module that does not admit an indecomposable decomposition.  ...  All facts from the model theory of modules over a uniserial ring cited below can be found in [4] and [12] .  ...  The machinery used is a mixture of general ring theory and model theory of modules, where we try to combine the advantages of each approach.  ... 
doi:10.1112/s0024610701002344 fatcat:yoiqftgktzaidkrumpjzohokby

Page 5895 of Mathematical Reviews Vol. , Issue 96j [page]

1996 Mathematical Reviews  
Puninskii], £-pure-injective modules over serial rings (145- 162); Theodore G. Faticoni, Modules over endomorphism rings as homotopy classes (163-183); Alexander A. Fomin [A. A. Fomin] and William J.  ...  Smith, Dual relative Krull dimension of modules over commutative rings (1-15); Pham Ngoc Anh [Pham Ngoc Anh], Morita duality, linear compactness and AB5*: a survey (17-28); D. M.  ... 

Artinian Serial Rings [chapter]

Gennadi Puninski
2001 Serial Rings  
So the theory of modules over artinian serial rings seems to be completely clear. Slightly less is known on the structure of artinian serial rings itselves.  ...  Also every serial group ring of a finite group over a field is QF.  ...  .] , [17, Thm. 3.3 ] every indecomposable right module over R is a homomorphic image of a module e i R. Thus we may assume that e 1 R is faithful.  ... 
doi:10.1007/978-94-010-0652-1_8 fatcat:7bbp53kl6bd7tek2vrfyojyafq

Page 16 of Mathematical Reviews Vol. , Issue 94i [page]

1994 Mathematical Reviews  
The aim of the representation theory of a finite-dimensional alge- bra A is to understand the category mod A of finite-dimensional modules over A.  ...  nilpotency index, let A be a left finitely generated module over R and let S be the endomorphism ring of A.  ... 
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