Prime submodules of Noetherian modules. (English) Zbl 0814.16017

McCasland, R. L.; Smith, P. F.

Rocky Mt. J. Math. 23, No. 3, 1041-1062 (1993).

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.

Reviewer: Dinh van Huynh (Pusan)

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

Keywords:

annihilator; left Goldie; prime ideals; chain conditions; prime submodules; finitely generated left \(R\)-module; strongly prime submodules

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