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Prime submodules of Noetherian modules. (English) Zbl 0814.16017
Let $$M$$ be a left module over a ring $$R$$. Then a proper submodule $$N$$ of $$M$$ is defined to be a prime submodule if for each $$r \in R$$, $$m \in M$$, $$rRm \subseteq N$$ implies $$rM \subseteq N$$ or $$m \in N$$. Hence if $$N$$ is a prime submodule of $$M$$, then the annihilator $$P$$ of $$M/N$$ is a two- sided prime ideal of $$R$$. A proper submodule $$K$$ of $$M$$ is called a strongly prime submodule of $$M$$ if the annihilator $$Q$$ of $$M/K$$ satisfies the following conditions: (i) $$Q$$ is a prime ideal of $$R$$ and $$R/Q$$ is left Goldie; (ii) $$M/K$$ is a torsion-free left $$(R/Q)$$-module. A strongly prime submodule of $$M$$ is prime. The authors study the question, which prime ideals $$P$$ (resp. $$Q$$) of $$R$$ are the annihilators of $$M/N$$ with $$N$$ prime (strongly prime) in $$M$$. They also obtain interesting results on chain conditions for (strongly) prime submodules, for example, it is shown that if a ring $$R$$ has ACC (resp. DCC) on prime ideals then any finitely generated left $$R$$-module satisfies ACC (resp., DCC) on strongly prime submodules.

##### MSC:
 16P40 Noetherian rings and modules (associative rings and algebras) 16P60 Chain conditions on annihilators and summands: Goldie-type conditions 16D25 Ideals in associative algebras 16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
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