Properties of top modules.(English)Zbl 1136.16009

The notion of Zariski topology on the set of all prime ideals of an associative ring with identity is transferred to the case of modules in the following way: A proper submodule $$P$$ of right $$R$$-module $$M$$ is called ‘prime’ provided $$mRr\subset P$$, for $$m\in M$$ and $$r\in R$$ implies $$m\in P$$ or $$Mr\subset P$$. For $$N\leqslant M$$ by $$U(N)$$ is denoted the set $$\{P:P$$ is prime and $$P\nsupseteq N\}$$. A module $$M$$ is called a ‘right top module’ provided the family $$\{U(N)\}_{N\leqslant M}$$ is closed under taking finite intersections.
Properties of right top modules were studied in several papers [R. L. McCasland, M. E. Moore and P. F. Smith, Commun. Algebra 25, No. 1, 79-103 (1997; Zbl 0876.13002); G. Zhang, J. Nanjing Univ., Math. Biq. 16, No. 1, 42-52 (1999; Zbl 0978.16004); G. Zhang and W. Tong, ibid. 17, No. 1, 15-20 (2000; Zbl 0976.16008)].
The author studies in the paper under review relations between the property of a module to be a right top module and other properties. It is proved, for instance, that a semisimple right $$R$$-module $$M$$ is a right top module if and only if $$M$$ is distributive. A right $$R$$-module $$M$$ over an Artinian semisimple ring is a right top module if and only if the module $$MI$$ is uniserial for every ideal $$I$$ of $$R$$ which is a simple ring.

MSC:

 16D80 Other classes of modules and ideals in associative algebras 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras 16S60 Associative rings of functions, subdirect products, sheaves of rings 16W80 Topological and ordered rings and modules 54H13 Topological fields, rings, etc. (topological aspects)