The small condition for modules with Noetherian dimension

A module M with Noetherian dimension is said to satisfy the small condition, if for any small submodule S of M the Noetherian dimension of S is strictly less than the Noetherian dimension of M. For an Artinian  module M, this is equivalent to that M is semisimple. In this article, we introduce  and study this concept and observe some basic facts for modules with this condition. As a main result, it is shown that if M is a  module with  finite hollow dimension which satisfies the  small condition, then \alpha \leq n-dim\, M\leq \alpha+۱, where  \alpha=\sup\{ n-dim\,S: S\ll M\}. Furthermore, if M is a  module with Krull dimension and finite hollow dimension, then \alpha \leq k-dim\, M\leq \alpha+۱, where  \alpha=\sup\{ k-dim\,S: S\ll M\}.  Also, we study the projective cover of modules satisfying the small condition or with finite hollow dimension

کلیدواژه ها:

Large condition ، Small condition ، Semiatomic modules ، Projective cover