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Hereditary CS-modules. (English) Zbl 0798.16005
The starting point of this paper is a result of B. L. Osofsky [Pac. J. Math. 14, 645-650 (1964; Zbl 0145.266)] which asserts that a right hereditary right self-injective ring is semisimple. The authors extend this result from rings to modules. A module $$M$$ is called hereditary if every submodule of $$M$$ is projective and $$M$$ is a CS-module when every submodule of $$M$$ is essential in a direct summand of $$M$$. The main result is that a hereditary CS-module is a direct sum of noetherian uniform modules. The paper ends with several applications. For instance, it is shown that, for any ring $$R$$ that is either commutative or semiprime, any hereditary continuous right $$R$$-module is semisimple. Another consequence is that any right continuous ring with all right ideals countably generated is semiperfect, from which it follows that, for any ring $$R$$, any countable continuous right $$R$$-module is a direct sum of uniform modules.

##### MSC:
 16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) 16E60 Semihereditary and hereditary rings, free ideal rings, Sylvester rings, etc. 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras 16D50 Injective modules, self-injective associative rings 16P40 Noetherian rings and modules (associative rings and algebras)
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