In this article,we introduce and study the concept of countably generated dimension,which is a Krull-like dimension extension of the concept of DCC on countably generated submodules.We show that some of the basic resu...In this article,we introduce and study the concept of countably generated dimension,which is a Krull-like dimension extension of the concept of DCC on countably generated submodules.We show that some of the basic results of Krull dimension are true for countably generated dimension.It is shown that an jR-module M has Krull dimension if and only if it has countably generated dimension,and its Krull dimension and countably generated dimension coincide.展开更多
Given a significative class F of commutative rings, we study the precise conditions under which a commutative ring R has an S-envelope. A full answer is obtained when F is the class of fields, semisimple commutative r...Given a significative class F of commutative rings, we study the precise conditions under which a commutative ring R has an S-envelope. A full answer is obtained when F is the class of fields, semisimple commutative rings or integral domains. When .F is the class of Noetherian rings, we give a full answer when the Krull dimension of R is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.展开更多
Let D be a finite-dimensional integral domain, Spec（D） be the set of prime ideals of D, and SpSS（D） be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structur...Let D be a finite-dimensional integral domain, Spec（D） be the set of prime ideals of D, and SpSS（D） be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structure of D with ｜SpSS（D）｜ = n ＋ dim（D） for 1 ≤ n ≤5 except for the quasi-local cases of n = 4, 5. In this paper, we show that there is an integral domain D such that ｜SpSS（D） ｜ = n＋dim（D） for all positive integers n with n ≠ 2. As corollaries, we completely characterize the quasi-local domains D with ｜SpSS（D）｜= n＋dim（D） for n = 4, 5. Furthermore, we also present the lower and upper bounds of ISpSS（D）I when Spee（D） is a finite tree.展开更多
基金The author is grateful to the Research Council of Shahid Chamran University of Ahvaz for financial support(SCU.MM99.192).
文摘In this article,we introduce and study the concept of countably generated dimension,which is a Krull-like dimension extension of the concept of DCC on countably generated submodules.We show that some of the basic results of Krull dimension are true for countably generated dimension.It is shown that an jR-module M has Krull dimension if and only if it has countably generated dimension,and its Krull dimension and countably generated dimension coincide.
基金supported by research projects from the Fundación ‘Sneca’ of Murcia (Programa de Ayudas a Grupos de Excelencia)the Spanish Ministry of Science and Innovation (Programa Nacional de Proyectos de Investigación Fundamental), with a part of FEDER funds
文摘Given a significative class F of commutative rings, we study the precise conditions under which a commutative ring R has an S-envelope. A full answer is obtained when F is the class of fields, semisimple commutative rings or integral domains. When .F is the class of Noetherian rings, we give a full answer when the Krull dimension of R is zero and when the envelope is required to be epimorphic. The general problem is reduced to identifying the class of non-Noetherian rings having a monomorphic Noetherian envelope, which we conjecture is the empty class.
文摘Let D be a finite-dimensional integral domain, Spec（D） be the set of prime ideals of D, and SpSS（D） be the set of spectral semistar operations on D. Mimouni gave a complete description for the prime ideal structure of D with ｜SpSS（D）｜ = n ＋ dim（D） for 1 ≤ n ≤5 except for the quasi-local cases of n = 4, 5. In this paper, we show that there is an integral domain D such that ｜SpSS（D） ｜ = n＋dim（D） for all positive integers n with n ≠ 2. As corollaries, we completely characterize the quasi-local domains D with ｜SpSS（D）｜= n＋dim（D） for n = 4, 5. Furthermore, we also present the lower and upper bounds of ISpSS（D）I when Spee（D） is a finite tree.
