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Paul C. Eklof, Saharon Shelah
<span title="2000-11-27">2000</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
As before, let M = ϕ[F ], M α = ϕ[F α ], M α,τ = ϕ[F α,τ ] and let C be a club such that for α ∈ C, M α ⊆ A α . Fix δ 1 in C ∩ S. Let δ be δ + 1 and choose γ ∈ C such that γ > δ.  ...  There is a club C ′ in µ such that for ν ∈ C ′ , M δ 1 ,ν ⊆ B γ,ν and A δ+1 ∩ B γ,ν ⊆ B δ+1,ν .  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0011228v1">arXiv:math/0011228v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/cm5ctern6bbobamj3tsbkl4cyi">fatcat:cm5ctern6bbobamj3tsbkl4cyi</a> </span>
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### Explicitly nonstandard uniserial modules [article]

Paul C. Eklof, Saharon Shelah
<span title="1993-01-15">1993</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
σ , c ρ ∈ R * satisfy c ρ ≡ c σ e ρ σ (mod r σ ), then f σ (c σ ) < f ρ (c ρ ).  ...  ,S 1 ≤S 2 if and only if there is a cub C such that S 1 ∩ C ⊆ S 2 ∩ C.  ...  We first define, by induction on σ, f σ (c σ ) -or, more precisely, f σ (c σ + r σ R) -for all c σ ∈ R * such that c σ ≡ z 0 e σ 0 (mod r 0 ).  ...
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### A non-reflexive Whitehead group [article]

Paul C. Eklof, Saharon Shelah
<span title="1999-08-29">1999</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
y) = t ∧ḣ(y + K) = c ℓ for ℓ = 1, 2, then c 1 = c 2 . which is a contradiction of the choice of m.  ...  Let p * denote the "heart" of {p δ : δ ∈ S ′ }; that is, dom(p * ) = C and for all µ ∈ C, dom(p * (µ)) = dom(p δ 1 (µ)) ∩ dom(p δ 2 (µ)) (= C µ , say) for δ 1 = δ 2 ∈ S ′ ; and p * (µ) = p δ1 (µ) ↾ C µ  ...  1 − c 2  ...
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### Modules with Strange Decomposition Properties [chapter]

Paul C. Eklof
<span title="">2000</span> <i title="Birkhäuser Basel"> Infinite Length Modules </i> &nbsp;
II Are A; B 2 C isomorphic whenever A 2 is isomorphic to B 2 ?  ...  answers to the following two Kaplansky test problems cf. 22, pp. 12f : I Are A; B 2 C isomorphic whenever A is isomorphic to a direct summand of B and B is isomorphic to a direct summand of A?  ...  g will be in C . For any 1 ; :::; n in End R M, C 1 ::: C n is a club so E C 1 ::: C n is non-empty and we can choose in C 1 ::: C n so that also 2 E.  ...
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### Absolutely rigid systems and absolutely indecomposable groups [article]

Paul C. Eklof, Saharon Shelah
<span title="2000-10-27">2000</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Φ ν (σ ↾ k) ⊆ Φ µ (θ(σ ↾ k)) = Φ µ ( f (0), ..., f (k − 1) ) every quantifier-free formula satisfied by a, b, c in A ν (e.g. a = 0, a − b = c, ab = c) is satisfied by f (a), f (b), f (c) in A µ .  ...  Thus an atomic formula is one of the form n i=0 c i x i = 0 where the c i are integers and the x i are variables.  ...  c)a α z↾m−1 , which is impossible unless c = c ′ .  ...
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Paul C Eklof, Saharon Shelah
<span title="">1991</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/h7qx4qsc2zf7hiupv27dke5fk4" style="color: black;">Journal of Algebra</a> </i> &nbsp;
EKLOF AND SHELAH (b) P=P, * Paa, where, in VPs, P,, is the direct limit of P;1 (/I E Ord), with Pz the Easton support iteration (P,*, 0: : j< 8, i < fi) where &=o,+i.  ...  EKLOF AND SHELAH (We could, in fact, prove that MT is not the direct summand of an F-free module.) It suflices to prove that for 6 E E {r > 6 : AT/A8 is not F-free} is stationary in 1.  ...
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### Test Groups for Whitehead Groups [article]

Paul C. Eklof and Laszlo Fuchs and Saharon Shelah
<span title="2007-02-11">2007</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Notice that if C is a non-zero separable torsion-free group, then Ext(A, C) = 0 implies A is a W-group (since Z is a summand of C), but the converse may not hold, that is, Ext(A, C) may be non-zero for  ...  We are looking for groups C other than Z such that a group A is a W-group if and only if Ext(A, C) = 0; such a C will be called a test group for Whitehead groups, or a W-test group for short.  ...
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/math/0702293v1">arXiv:math/0702293v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/no43kn3tmjf3piwonlv6535jpu">fatcat:no43kn3tmjf3piwonlv6535jpu</a> </span>
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### On invariants for omega_1-separable groups [article]

Paul C. Eklof, Matthew Foreman, Saharon Shelah
<span title="1995-01-15">1995</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let C δ,n = K βn,βn+1 .  ...  Lemma 1 Suppose A ⊆ B and A ′ ⊆ B ′ ⊆ C ′ where C ′ /B ′ is ℵ 1 -free, B/A is countable and (B/A) * = 0. If θ : B → C ′ such that θ[A] ⊆ A ′ , then θ[B] ⊆ B ′ .  ...
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### New non-free Whitehead groups (corrected version) [article]

Paul C. Eklof, Saharon Shelah
<span title="1997-11-15">1997</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let p * denote the "heart" of the ∆-system; that is, dom(p * ) = C and for all µ ∈ C, dom(p * (µ)) = dom(p ξ 1 (µ)) ∩ dom(p ξ 2 (µ)) (= C µ , say) for ξ 1 = ξ 2 ∈ S; and p * (µ) ↾ C µ = p ξ 1 (µ) ↾ C µ  ...  Moreover, for every j ∈ C, {dom(p ξ (j)) : ξ ∈ S} forms a ∆-system and for all ξ 1 = ξ 2 in S, p ξ 1 (j) and p ξ 2 (j) agree on dom(p ξ 1 (j)) ∩ dom(p ξ 2 (j)).  ...
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### Whitehead modules over large principal ideal domains [article]

Paul C. Eklof, Saharon Shelah
<span title="2000-11-27">2000</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Thus b = p n c for some c ∈ R ′ and n ≥ m. But then b t = p n c t ∈ p m R ′ (p) . Therefore I = p m R ′ (p) . Proof of Theorem 1. Let λ = 2 ℵ1 .  ...
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### On a conjecture regarding nonstandard uniserial modules [article]

Paul C. Eklof, Saharon Shelah
<span title="1993-08-15">1993</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We define the c σ by induction. Let c 0 = 1; if c σ has been defined, let c σ+1 = e σ+1 σ c σ + X (σ,i) .  ...  Then we have c δ ≡ e δ σ c σ ≡ u σẽ δ σ c σ and d δ ≡ d σẽ δ σ , so f ≡ u −1 σ c −1 σ d σ .  ...
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### Torsion modules, lattices and p-points [article]

Paul C. Eklof, Birge Huisgen--Zimmermann, Saharon Shelah
<span title="1997-03-15">1997</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The claim concerning a realization of L ′ in case L is the dual ideal lattice of a domain, is an immediate consequence of part (c) of Corollary C of [8] . 2 proof of Theorem 5.  ...  By Corollary C of [8] , there exists a complete upper subsemilattice L ′ of L which is isomorphic to an upper semilattice of the form L(U ) for some ultrafilter U on ω.  ...
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### On the cogeneration of cotorsion pairs [article]

Paul C. Eklof and Saharon Shelah and Jan Trlifaj
<span title="2004-05-07">2004</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Enochs' solution of the Flat Cover Conjecture was extended as follows: (*) If C is a cotorsion pair generated by a class of cotorsion modules, then C is cogenerated by a set.  ...  We show that (*) is the best result provable in ZFC in case R has a countable spectrum: the Uniformization Principle UP^+ implies that C is not cogenerated by a set whenever C is a cotorsion pair generated  ...  Introduction For any ring R, if S is a class of (right) Rmodules, we define A cotorsion pair (originally called a cotorsion theory) is a pair C = (F , C) such that F = ⊥ C and C = F ⊥ .  ...
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### The Kaplansky test problems for aleph_1-separable groups [article]

Paul C. Eklof, Saharon Shelah
<span title="1997-09-15">1997</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Since ng ∈ G n−1 , ng = Σ i∈I e i,n−1 c i + c 0 for some c i , c 0 ∈ A. Then ng = Σ i∈I (ne i,n − a i )c i + c 0 = nΣ i∈I e i,n c i + a ′ for some a ′ ∈ A.  ...  Suppose that Σ i∈I e i,n c i + 1 · c 0 = 0 for some c 0 , c i ∈ A. Then Σ i∈I ne i,n c i + nc 0 = 0, so Σ i∈I e i,n−1 c i + 1 · (Σ i∈I a i c i + nc 0 ) = 0.  ...
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### Covers Induced by Ext

Paul C Eklof, Jan Trlifaj
<span title="">2000</span> <i title="Elsevier BV"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/h7qx4qsc2zf7hiupv27dke5fk4" style="color: black;">Journal of Algebra</a> </i> &nbsp;
of C; (c) ⊥ C = ⊥ F, where F is the flat cover of C; moreover, F is pure- injective.  ...  We define ⊥ = Ker Ext − = D Ext D C = 0 for all C ∈ and similarly ⊥ = Ker Ext − = D Ext C D = 0 for all C ∈ Ker Tor − = A Tor A B = 0 for all B ∈ For a module C, we will write ⊥ C instead of ⊥ C We start  ...
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