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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
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