On the prime spectrum of \(X\)-injective modules. (English) Zbl 1197.13012
Let \(M\) be a unitary \(R\)-module, \(R\) a commutative ring with \(1 \neq 0\). \(M\) is said to be \(X\)-injective, where \(X=\mathrm{Spec}(M)\) if either \(X= \phi\) or \(X \neq \phi\) and the natural map of \(X\) is injective. It is shown that every top module and every weak multiplication module are \(X\)-injective, but by means of examples the converse is shown to be false. In the case of a finitely generated module \(M\) these notions do coincide, as does that of a multiplication module.
Characterizations of \(X\)-injective modules are found; for example, \(M\) is an \(X\)-injective \(R\)-module if and only if \(M_{p}\) is an \(X\)-injective \(R_{p}\)-module for every prime ideal \(p\) of \(R\). \(X\)-injective modules are characterized in some special cases. For example, in the case where \(M\) is a free \(R\)-module, or where \(R\) is a perfect ring, it is proven that \(M\) is \(X\)-injective if and only if \(M\) is cyclic, if and only if \(M\) is a top module. For the case where \(M\) is a projective \(R\)-module, \(M\) is \(X\)-injective if and only if \(M\) is locally cyclic, if and only if \(M\) is a top module.
A domain over which every \(X\)-injective \(R\)-module is cyclic, is shown to be a field. If \(R\) is a PID, then an \(R\)-module \(M\) is proven to be a weak multiplication module if and only if \(M\) is a top module which is torsion or torsion-free.
In the final section, primeful \(R\)-modules are considered, i.e. \(R\)-modules such that either \(M=(0)\) or \(M \neq (0)\) and the natural map of \(X\) is surjective. \(\tau\) denotes the Zariski topology on \(X\) and \(\tau ^{*}\) the quasi-Zariski topology. A topological space \(T\) is called a spectral space if \(T\) is homeomorphic to \(\mathrm{Spec}(S)\) with the Zariski topology for some ring \(S\). A nonzero primeful top \(Z\)-module \(M\) is introduced for which \((X,\tau )\) is a spectral space, but \((X,\tau ^{*})\) is not spectral. Sufficient conditions for \((X,\tau ^{*})\) to be a spectral space, for the case where \(M\) is a primeful top \(R\)-module, are subsequently found.
Characterizations of \(X\)-injective modules are found; for example, \(M\) is an \(X\)-injective \(R\)-module if and only if \(M_{p}\) is an \(X\)-injective \(R_{p}\)-module for every prime ideal \(p\) of \(R\). \(X\)-injective modules are characterized in some special cases. For example, in the case where \(M\) is a free \(R\)-module, or where \(R\) is a perfect ring, it is proven that \(M\) is \(X\)-injective if and only if \(M\) is cyclic, if and only if \(M\) is a top module. For the case where \(M\) is a projective \(R\)-module, \(M\) is \(X\)-injective if and only if \(M\) is locally cyclic, if and only if \(M\) is a top module.
A domain over which every \(X\)-injective \(R\)-module is cyclic, is shown to be a field. If \(R\) is a PID, then an \(R\)-module \(M\) is proven to be a weak multiplication module if and only if \(M\) is a top module which is torsion or torsion-free.
In the final section, primeful \(R\)-modules are considered, i.e. \(R\)-modules such that either \(M=(0)\) or \(M \neq (0)\) and the natural map of \(X\) is surjective. \(\tau\) denotes the Zariski topology on \(X\) and \(\tau ^{*}\) the quasi-Zariski topology. A topological space \(T\) is called a spectral space if \(T\) is homeomorphic to \(\mathrm{Spec}(S)\) with the Zariski topology for some ring \(S\). A nonzero primeful top \(Z\)-module \(M\) is introduced for which \((X,\tau )\) is a spectral space, but \((X,\tau ^{*})\) is not spectral. Sufficient conditions for \((X,\tau ^{*})\) to be a spectral space, for the case where \(M\) is a primeful top \(R\)-module, are subsequently found.
Reviewer: Frieda Theron (Spruce Grove, Alberta)
MSC:
| 13C13 | Other special types of modules and ideals in commutative rings |
| 13C99 | Theory of modules and ideals in commutative rings |
Keywords:
\(X\)-injective module; prime submodule; spectral space; weak multiplication module, Zariski topology; quasi-Zariski topologyReferences:
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