Mathematical Logic Quarterly最新文献

{"title":"Normalizing notations in the Ershov hierarchy","authors":"Cheng Peng","doi":"10.1002/malq.202100004","DOIUrl":"https://doi.org/10.1002/malq.202100004","url":null,"abstract":"<p>The Turing degrees of infinite levels of the Ershov hierarchy were studied by Liu and Peng [8]. In this paper, we continue the study of Turing degrees of infinite levels and lift the study of density property to the levels beyond ω². In doing so, we rely on notations with some nice properties. We introduce the concept of normalizing notations and generate normalizing notations for higher levels. The generalizations of the weak density theorem and the nondensity theorem are proved for higher levels in the Ershov hierarchy. Furthermore, we also investigate the minimal degrees in the infinite levels of the Ershov hierarchy.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"67 4","pages":"506-513"},"PeriodicalIF":0.3,"publicationDate":"2021-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72362540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"MA\u0000 (\u0000 \u0000 ℵ\u0000 0\u0000 \u0000 )\u0000 restricted to complete Boolean algebras and choice","authors":"Eleftherios Tachtsis","doi":"10.1002/malq.202000031","DOIUrl":"https://doi.org/10.1002/malq.202000031","url":null,"abstract":"<p>It is a long standing open problem whether or not the Axiom of Countable Choice implies the fragment <math>\u0000 <mrow>\u0000 <mi>MA</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow></math> of Martin's Axiom either in <math>\u0000 <mi>ZFA</mi></math> or in <math>\u0000 <mi>ZF</mi></math>. In this direction, we provide a partial answer by establishing that the Boolean Prime Ideal Theorem in conjunction with the Countable Union Theorem does not imply <math>\u0000 <mrow>\u0000 <mi>MA</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow></math> restricted to complete Boolean algebras in <math>\u0000 <mi>ZF</mi></math>. Furthermore, we prove that the latter (formally) weaker form of <math>\u0000 <mrow>\u0000 <mi>MA</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow></math> and the Δ-system Lemma are independent of each other in <math>\u0000 <mi>ZFA</mi></math>.</p><p>We also answer open questions from Tachtsis [16] which concern the status of <math>\u0000 <mrow>\u0000 <mi>MA</mi>\u0000 <mo>(</mo>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>)</mo>\u0000 </mrow></math> restricted to complete Boolean algebras in certain Fraenkel–Mostowski permutation models of <math>\u0000 <mi>ZFA</mi></math> and we strengthen some results from the above paper.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"67 4","pages":"420-431"},"PeriodicalIF":0.3,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72361979","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Effective aspects of Jacobson radicals of rings","authors":"Huishan Wu","doi":"10.1002/malq.202100002","DOIUrl":"https://doi.org/10.1002/malq.202100002","url":null,"abstract":"<p>This paper studies effective aspects of Jacobson radicals of rings and their applications from the viewpoint of reverse mathematics. First, we propose four radicals of rings, showing that the first order (resp., second order) left and right Jacobson radical coincide in <math>\u0000 <msub>\u0000 <mi>RCA</mi>\u0000 <mn>0</mn>\u0000 </msub></math> (resp., <math>\u0000 <msub>\u0000 <mi>ACA</mi>\u0000 <mn>0</mn>\u0000 </msub></math>). Second, we study Jacobson radicals in left (resp., right) local rings and show that the second order left and right Jacobson radical of left (resp., right) local rings coincide within <math>\u0000 <msub>\u0000 <mi>RCA</mi>\u0000 <mn>0</mn>\u0000 </msub></math>. Third, we apply our results about Jacobson radicals of rings and local rings to study properties of left (resp., right) strongly indecomposable modules; furthermore, we study effective aspects of typical lemmas and theorems related to strongly indecomposable modules, and show that <math>\u0000 <msub>\u0000 <mi>RCA</mi>\u0000 <mn>0</mn>\u0000 </msub></math> proves the Fitting Lemma as well as the Krull-Schmidt-Azumaya Theorem and that <math>\u0000 <msub>\u0000 <mi>ACA</mi>\u0000 <mn>0</mn>\u0000 </msub></math> proves the Krull-Schmidt Theorem.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"67 4","pages":"489-505"},"PeriodicalIF":0.3,"publicationDate":"2021-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72361971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"When is a real generic over L?","authors":"Fabiana Castiblanco,&nbsp;Ralf Schindler","doi":"10.1002/malq.202100012","DOIUrl":"https://doi.org/10.1002/malq.202100012","url":null,"abstract":"<p>In this paper we isolate a new abstract criterion for when a given real <i>x</i> is generic over <i>L</i> in terms of being able to lift elementary embeddings of initial segments of <i>L</i> to <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 <mo>[</mo>\u0000 <mi>x</mi>\u0000 <mo>]</mo>\u0000 </mrow>\u0000 <annotation>$L[x]$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"27-31"},"PeriodicalIF":0.3,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202100012","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134806552","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Weakly binary expansions of dense meet-trees","authors":"Rosario Mennuni","doi":"10.1002/malq.202000045","DOIUrl":"10.1002/malq.202000045","url":null,"abstract":"<p>We compute the domination monoid in the theory <math>\u0000 <semantics>\u0000 <mi>DMT</mi>\u0000 <annotation>$mathsf {DMT}$</annotation>\u0000 </semantics></math> of dense meet-trees. In order to show that this monoid is well-defined, we prove <i>weak binarity</i> of <math>\u0000 <semantics>\u0000 <mi>DMT</mi>\u0000 <annotation>$mathsf {DMT}$</annotation>\u0000 </semantics></math> and, more generally, of certain expansions of it by binary relations on sets of open cones, a special case being the theory <math>\u0000 <semantics>\u0000 <mi>DTR</mi>\u0000 <annotation>$mathsf {DTR}$</annotation>\u0000 </semantics></math> from [7]. We then describe the domination monoids of such expansions in terms of those of the expanding relations.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"32-47"},"PeriodicalIF":0.3,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202000045","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85493010","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"ℵ0-categorical Banach spaces contain \u0000 \u0000 ℓ\u0000 p\u0000 or c0","authors":"Karim Khanaki","doi":"10.1002/malq.201800086","DOIUrl":"https://doi.org/10.1002/malq.201800086","url":null,"abstract":"<p>This paper has three parts. First, we establish some of the basic model theoretic facts about <math>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mi>T</mi>\u0000 </msub></math>, the Tsirelson space of Figiel and Johnson [20]. Second, using the results of the first part, we give some facts about general Banach spaces. Third, we study model-theoretic dividing lines in some Banach spaces and their theories. In particular, we show: (1) <math>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mi>T</mi>\u0000 </msub></math> has the <i>non independence property</i>\u0000(NIP); (2) every Banach space that is ℵ₀-categorical up to small perturbations embeds <i>c</i>₀ or <math>\u0000 <msub>\u0000 <mi>ℓ</mi>\u0000 <mi>p</mi>\u0000 </msub></math> (<math>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>⩽</mo>\u0000 <mi>p</mi>\u0000 <mo>&lt;</mo>\u0000 <mi>∞</mi>\u0000 </mrow></math>) almost isometrically; consequently the (continuous) first-order theory of <math>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mi>T</mi>\u0000 </msub></math> does not characterize <math>\u0000 <msub>\u0000 <mi>M</mi>\u0000 <mi>T</mi>\u0000 </msub></math>, up to almost isometric isomorphism.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"67 4","pages":"469-488"},"PeriodicalIF":0.3,"publicationDate":"2021-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72318202","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The profinite topology of free groups and weakly generic tuples of automorphisms","authors":"Gábor Sági","doi":"10.1002/malq.202000066","DOIUrl":"https://doi.org/10.1002/malq.202000066","url":null,"abstract":"&lt;p&gt;Let &lt;math&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt; be a countable first order structure and endow the universe of &lt;math&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt; with the discrete topology. Then the automorphism group &lt;math&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Aut&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; of &lt;math&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;&lt;/math&gt; becomes a topological group. A tuple of automorphisms &lt;math&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⟨&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;…&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;⟩&lt;/mo&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; is defined to be weakly generic iff its diagonal conjugacy class (in the algebraic sense) is dense (in the topological sense) and the &lt;math&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;⟨&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mn&gt;0&lt;/mn&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;mi&gt;…&lt;/mi&gt;\u0000 &lt;mo&gt;,&lt;/mo&gt;\u0000 &lt;msub&gt;\u0000 &lt;mi&gt;g&lt;/mi&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;n&lt;/mi&gt;\u0000 &lt;mo&gt;−&lt;/mo&gt;\u0000 &lt;mn&gt;1&lt;/mn&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;/msub&gt;\u0000 &lt;mo&gt;⟩&lt;/mo&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt;-orbit of each &lt;math&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;a&lt;/mi&gt;\u0000 &lt;mo&gt;∈&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; is finite. Existence of tuples of weakly generic automorphisms are interesting from the point of view of model theory as well as from the point of view of finite combinatorics.&lt;/p&gt;&lt;p&gt;The main results of the present work are as follows. In Theorem 2.6 we characterize the existence of tuples of weakly generic automorphisms with the aid of the profinite topology of free groups. In Corollary 2.12 we will show that if &lt;math&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Aut&lt;/mi&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; has finite topological rank &lt;i&gt;r&lt;/i&gt; (and satisfies a further, mild technical condition) then the existence of a weakly generic tuple in &lt;math&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mi&gt;Aut&lt;/mi&gt;\u0000 &lt;msup&gt;\u0000 &lt;mrow&gt;\u0000 &lt;mo&gt;(&lt;/mo&gt;\u0000 &lt;mi&gt;A&lt;/mi&gt;\u0000 &lt;mo&gt;)&lt;/mo&gt;\u0000 &lt;/mrow&gt;\u0000 &lt;mi&gt;r&lt;/mi&gt;\u0000 &lt;/msup&gt;\u0000 &lt;/mrow&gt;&lt;/math&gt; implies the existence of weakly ","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"67 4","pages":"432-444"},"PeriodicalIF":0.3,"publicationDate":"2021-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/malq.202000066","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72314246","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Generic expansion of an abelian variety by a subgroup","authors":"Christian d'Elbée","doi":"10.1002/malq.202000017","DOIUrl":"https://doi.org/10.1002/malq.202000017","url":null,"abstract":"<p>Let <i>A</i> be an abelian variety in an algebraically closed field of characteristic 0. We prove that the expansion of <i>A</i> by a generic divisible subgroup of <i>A</i> with the same torsion exists provided <i>A</i> has few algebraic endomorphisms, namely <math>\u0000 <mrow>\u0000 <mi>End</mi>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mi>Z</mi>\u0000 </mrow></math>. The resulting theory is NSOP₁ and not simple. Note that there exist abelian varieties <i>A</i> with <math>\u0000 <mrow>\u0000 <mi>End</mi>\u0000 <mo>(</mo>\u0000 <mi>A</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mi>Z</mi>\u0000 </mrow></math> of any genus.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"67 4","pages":"402-408"},"PeriodicalIF":0.3,"publicationDate":"2021-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72351942","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Remarks on infinite factorials and cardinal subtraction in \u0000 \u0000 ZF\u0000 $mathsf{ZF}$","authors":"Guozhen Shen","doi":"10.1002/malq.202100034","DOIUrl":"10.1002/malq.202100034","url":null,"abstract":"<p>The factorial of a cardinal <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>, denoted by <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>!</mo>\u0000 </mrow>\u0000 <annotation>$mathfrak {a}!$</annotation>\u0000 </semantics></math>, is the cardinality of the set of all permutations of a set which is of cardinality <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>. We give a condition that makes the cardinal equality <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>!</mo>\u0000 <mo>−</mo>\u0000 <mi>a</mi>\u0000 <mo>=</mo>\u0000 <mi>a</mi>\u0000 <mo>!</mo>\u0000 </mrow>\u0000 <annotation>$mathfrak {a}!-mathfrak {a}=mathfrak {a}!$</annotation>\u0000 </semantics></math> provable without the axiom of choice. In fact, we prove in <math>\u0000 <semantics>\u0000 <mi>ZF</mi>\u0000 <annotation>$mathsf {ZF}$</annotation>\u0000 </semantics></math> that, for all cardinals <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>, if <math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mi>ℵ</mi>\u0000 <mn>0</mn>\u0000 </msub>\u0000 <mo>⩽</mo>\u0000 <msup>\u0000 <mn>2</mn>\u0000 <mi>a</mi>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$aleph _0leqslant 2^mathfrak {a}$</annotation>\u0000 </semantics></math> and there is a permutation without fixed points on a set which is of cardinality <math>\u0000 <semantics>\u0000 <mi>a</mi>\u0000 <annotation>$mathfrak {a}$</annotation>\u0000 </semantics></math>, then <math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 <mo>!</mo>\u0000 <mo>−</mo>\u0000 <mi>a</mi>\u0000 <mo>=</mo>\u0000 <mi>a</mi>\u0000 <mo>!</mo>\u0000 </mrow>\u0000 <annotation>$mathfrak {a}!-mathfrak {a}=mathfrak {a}!$</annotation>\u0000 </semantics></math>.</p>","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"68 1","pages":"67-73"},"PeriodicalIF":0.3,"publicationDate":"2021-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72881935","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Effective aspects of Jacobson radicals of rings","authors":"Huishan Wu","doi":"10.1002/malq.202100002","DOIUrl":"https://doi.org/10.1002/malq.202100002","url":null,"abstract":"This paper studies effective aspects of Jacobson radicals of rings and their applications from the viewpoint of reverse mathematics. First, we propose four radicals of rings, showing that the first order (resp., second order) left and right Jacobson radical coincide in RCA0 (resp., ACA0 ). Second, we study Jacobson radicals in left (resp., right) local rings and show that the second order left and right Jacobson radical of left (resp., right) local rings coincide within RCA0 . Third, we apply our results about Jacobson radicals of rings and local rings to study properties of left (resp., right) strongly indecomposable modules; furthermore, we study effective aspects of typical lemmas and theorems related to strongly indecomposable modules, and show that RCA0 proves the Fitting Lemma as well as the Krull‐Schmidt‐Azumaya Theorem and that ACA0 proves the Krull‐Schmidt Theorem.","PeriodicalId":49864,"journal":{"name":"Mathematical Logic Quarterly","volume":"9 1","pages":""},"PeriodicalIF":0.3,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87168799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}