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Noetherian spectrum on rings and modules. (English) Zbl 1227.13009
A submodule $$N$$ of a finitely generated $$R$$-module $$M$$ is said to be primary if $$ax\in N$$, $$a\in R$$, $$x\in M \setminus N$$ implies $$a\in \mathrm{rad}(N :_R M)$$. If $$ax\in N$$, $$a\in R$$, $$x\in M \setminus N$$ implies $$a\in (N :_R M)$$, $$N$$ is said to be a prime submodule of $$M$$. It follows that if $$N$$ is primary then $$\mathrm{rad}(N :_R M)$$ is prime, and that a submodule $$N$$ is prime if and only if $$(N :_R M) = \mathfrak{p}$$ is prime ideal and $$M/N$$ is torsion-free over $$R/\mathfrak{p}$$. The set of prime submodules of $$M$$ is denoted by $$\mathrm{Spec}(M)$$. This paper continues with the research in the area of prime submodules. The author considers the properties of the prime ideals of a ring which have counterparts for the prime submodules of a module. It is shown that the well-known characterizations of when a commutative ring $$R$$ has Noetherian spectrum carry over to characterizations of when the set $$\mathrm{Spec}(M)$$ of prime submodules of a finitely generated module $$M$$ is Noetherian. The symmetric algebra $$S_R(M)$$ of $$M$$ is used to show that the Noetherian property of $$\mathrm{Spec}(R)$$, and some related properties, pass from the ring $$R$$ to the finitely generated $$R$$-modules.
The paper is well written and requires only familiarity with basic notions and so it is quite readable.
##### MSC:
 13C99 Theory of modules and ideals in commutative rings 13E15 Commutative rings and modules of finite generation or presentation; number of generators 13E99 Chain conditions, finiteness conditions in commutative ring theory
##### Keywords:
prime submodule; spectrum of module; Noethrian module
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