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Model theory of modules over a serial ring

1995
*
Annals of Pure and Applied Logic
*

We use the Drozd-Warfield structure theorem for finitely presented

doi:10.1016/0168-0072(94)00017-w
fatcat:akp5vvrukvcenmln7t5otdu2dm
*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. ...##
###
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

2003
*
Communications in Algebra
*

We prove that every

doi:10.1081/agb-120024862
fatcat:txvhtirgmvgg5oq5voktgqhk4q
*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*. ...##
###
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

1999
*
Bulletin of the London Mathematical Society
*

Ziegler [15] checked Vaught's conjecture for complete

doi:10.1112/s0024609398005360
fatcat:7p4jqxlv2fc4vjb3vxnm5y2hbu
*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. ...##
###
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

2001
*
Journal of the London Mathematical Society
*

We give an example

doi:10.1112/s0024610701002344
fatcat:yoiqftgktzaidkrumpjzohokby
*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. ...##
###
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]

2001
*
Serial Rings
*

So the

doi:10.1007/978-94-010-0652-1_8
fatcat:7bbp53kl6bd7tek2vrfyojyafq
*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. ...##
###
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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