Almost uniserial rings and modules

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• 作者：M. Behboodi^a ; ^b ; ^{mbehbood@cc.iut.ac.ir" class="auth_mail" title="E-mail the corresponding author} ; S. Roointan-Isfahani^a ; ^{s.roointan@math.iut.ac.ir" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 13A18 ; 13F30 ; 16D20 ; secondary ; 16D70 ; 16P20
• 刊名：Journal of Algebra
• 出版年：2016
• 出版时间：15 January 2016
• 年：2016
• 卷：446
• 期：Complete
• 页码：176-187
• 全文大小：332 K

文摘

We study the class of almost uniserial rings as a straightforward common generalization of left uniserial rings and left principal ideal domains. A ring R is called almost left uniserial if any two non-isomorphic left ideals of R are linearly ordered by inclusion, i.e., for every pair I, J of left ideals of R   either I⊆J$I\subseteq J$, or J⊆I$J\subseteq I$, or I≅J$I\cong J$. Also, an R-module M is called almost uniserial if any two non-isomorphic submodules are linearly ordered by inclusion. We give some interesting and useful properties of almost uniserial rings and modules. It is shown that a left almost uniserial ring is either a local ring or its maximal left ideals are cyclic. A Noetherian left almost uniserial ring is a local ring or a principal left ideal ring. Also, a left Artinian principal left ideal ring R is almost left uniserial if and only if R   is left uniserial or R=M₂(D)$R=M_2(D)$, where D is a division ring. Finally we consider Artinian commutative rings which are almost uniserial and we obtain a structure theorem for these rings.