Let H be a Krull monoid with infinite cyclic class group G that we identify with Z. Let GP ⊆ G denote the set of classes containing prime divisors. It was shown that ρ(H), the elasticity of H, is finite if and only if GP is bounded above or below. By a result of Lambert, it is easy to show that ρ(H) ≤ min{− inf(GP ), sup(GP )}. In this paper, we focus on H with large elasticity. When GP is infinite but bounded above or below, we give necessary and sufficient conditions for that ρ(H) > min{− inf(GP ), sup(GP )} − 3/2. When GP is a finite set, we give a better upper bound on ρ(H), which is sharp in some sense.
{"title":"Elasticities of Krull monoids with infinite cyclic class group","authors":"X. Zeng, Guixin Deng","doi":"10.1216/jca.2021.13.449","DOIUrl":"https://doi.org/10.1216/jca.2021.13.449","url":null,"abstract":"Let H be a Krull monoid with infinite cyclic class group G that we identify with Z. Let GP ⊆ G denote the set of classes containing prime divisors. It was shown that ρ(H), the elasticity of H, is finite if and only if GP is bounded above or below. By a result of Lambert, it is easy to show that ρ(H) ≤ min{− inf(GP ), sup(GP )}. In this paper, we focus on H with large elasticity. When GP is infinite but bounded above or below, we give necessary and sufficient conditions for that ρ(H) > min{− inf(GP ), sup(GP )} − 3/2. When GP is a finite set, we give a better upper bound on ρ(H), which is sharp in some sense.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"3 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84219892","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}
Pub Date : 2021-07-30DOI: 10.1007/978-3-030-89694-2_4
W. Bruns, A. Conca, M. Varbaro
{"title":"Castelnuovo–Mumford Regularity and Powers","authors":"W. Bruns, A. Conca, M. Varbaro","doi":"10.1007/978-3-030-89694-2_4","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_4","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"87 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84044406","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}
Pub Date : 2021-07-29DOI: 10.1007/978-3-030-89694-2_6
Laurent Bus'e, M. Chardin
{"title":"Fibers of Rational Maps and Elimination Matrices: An Application Oriented Approach","authors":"Laurent Bus'e, M. Chardin","doi":"10.1007/978-3-030-89694-2_6","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_6","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"6 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80307886","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}
Pub Date : 2021-06-28DOI: 10.1007/978-3-030-89694-2_7
Lars Christensen, Oana Veliche, J. Weyman
{"title":"Three Takes on Almost Complete Intersection Ideals of Grade 3","authors":"Lars Christensen, Oana Veliche, J. Weyman","doi":"10.1007/978-3-030-89694-2_7","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_7","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"8 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88370279","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}
Pub Date : 2021-06-28DOI: 10.1007/978-3-030-89694-2_5
G. Caviglia, Alessandro De Stefani, E. Sbarra
{"title":"The Eisenbud-Green-Harris Conjecture","authors":"G. Caviglia, Alessandro De Stefani, E. Sbarra","doi":"10.1007/978-3-030-89694-2_5","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_5","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"1 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82950669","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}
Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0rightarrow mathrm{Hom}_R(P,A)rightarrow mathrm{Hom}_R(P,B)rightarrow mathrm{Hom}_R(P,C)rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules, $u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.
{"title":"CHARACTERIZING S-PROJECTIVE MODULES AND S-SEMISIMPLE RINGS BY UNIFORMITY","authors":"Xiaolei Zhang, W. Qi","doi":"10.1216/jca.2023.15.139","DOIUrl":"https://doi.org/10.1216/jca.2023.15.139","url":null,"abstract":"Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $P$ is called uniformly $S$-projective provided that the induced sequence $0rightarrow mathrm{Hom}_R(P,A)rightarrow mathrm{Hom}_R(P,B)rightarrow mathrm{Hom}_R(P,C)rightarrow 0$ is $u$-$S$-exact for any $u$-$S$-short exact sequence $0rightarrow Arightarrow Brightarrow Crightarrow 0$. Some characterizations and properties of $u$-$S$-projective modules are obtained. The notion of $u$-$S$-semisimple modules is also introduced. A ring $R$ is called a $u$-$S$-semisimple ring provided that any free $R$-module is $u$-$S$-semisimple. Several characterizations of $u$-$S$-semisimple rings are provided in terms of $u$-$S$-semisimple modules, $u$-$S$-projective modules, $u$-$S$-injective modules and $u$-$S$-split $u$-$S$-exact sequences.","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"271 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73368745","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}
Pub Date : 2021-06-16DOI: 10.1007/978-3-030-89694-2_1
Josep Àlvarez Montaner, J. Jeffries, Luis N'unez-Betancourt
{"title":"Bernstein-Sato Polynomials in Commutative Algebra","authors":"Josep Àlvarez Montaner, J. Jeffries, Luis N'unez-Betancourt","doi":"10.1007/978-3-030-89694-2_1","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_1","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"226 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73948435","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}
Pub Date : 2021-06-15DOI: 10.1007/978-3-030-89694-2_13
Huy Tài Hà, N. Trung
{"title":"Depth Functions and Symbolic Depth Functions of Homogeneous Ideals","authors":"Huy Tài Hà, N. Trung","doi":"10.1007/978-3-030-89694-2_13","DOIUrl":"https://doi.org/10.1007/978-3-030-89694-2_13","url":null,"abstract":"","PeriodicalId":49037,"journal":{"name":"Journal of Commutative Algebra","volume":"27 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82320718","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}
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