کد مقاله کد نشریه سال انتشار مقاله انگلیسی نسخه تمام متن
8897269 1630736 2019 20 صفحه PDF دانلود رایگان
عنوان انگلیسی مقاله ISI
On the Schröder-Bernstein property for modules
کلمات کلیدی
موضوعات مرتبط
مهندسی و علوم پایه ریاضیات اعداد جبر و تئوری
پیش نمایش صفحه اول مقاله
On the Schröder-Bernstein property for modules
چکیده انگلیسی
The well known Schröder-Bernstein Theorem states that any two sets with one to one maps into each other are isomorphic. The question of whether any two (subisomorphic or) direct summand subisomorphic algebraic structures are isomorphic, has long been of interest. Kaplansky asked whether direct summands subisomorphic abelian groups are always isomorphic? The question generated a great deal of interest. The study of this question for the general class of modules has been somewhat limited. We extend the study of this question for modules in this paper. We say that a module Msatisfies the Schröder-Bernstein property (S-B property) if any two direct summands of M which are subisomorphic to direct summands of each other, are isomorphic. We show that a large number of classes of modules satisfy the S-B property. These include the classes of quasi-continuous, directly finite, quasi-discrete and modules with ACC on direct summands. It is also shown that over a Noetherian ring R, every extending module satisfies the S-B property. Among applications, it is proved that the class of rings R for which every R-module satisfies the S-B property is precisely that of pure-semisimple rings. We show that over a commutative domain R, any two quasi-continuous subisomorphic R-modules are isomorphic if and only if R is a PID. We study other conditions related to the S-B property and obtain characterizations of certain classes of rings via those conditions. Examples which delimit and illustrate our results are provided.
ناشر
Database: Elsevier - ScienceDirect (ساینس دایرکت)
Journal: Journal of Pure and Applied Algebra - Volume 223, Issue 1, January 2019, Pages 422-438
نویسندگان
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