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Interpretations of module varieties. (English) Zbl 0729.08005

For R any ring with unit, \({}_ R{\mathcal M}\) denote the variety of all R- modules. In this paper, the lattice \(L^{int}\) of varieties ordered by the relation of interpretability of varieties is studied, mainly applied to the varieties \({}_ R{\mathcal M}\). The main results are the following.
Theorem 1. There is a finitely presented commutative ring S such that \(_{{\mathbb{Z}}}{\mathcal M}<_ S{\mathcal M}\) and such that \({}_ S{\mathcal M}\leq_ R{\mathcal M}\) both for R any ring of algebraic numbers other than \({\mathbb{Z}}\) itself and for \(R={\mathbb{Z}}/n\) \((n=2,3,...)\). In fact, \({}_ S{\mathcal M}\) is equivalent to a finitely axiomatized variety in finitely many operation symbols.
Theorem 2. If \({\mathcal K}\) is a set of varieties, each \(>_{{\mathbb{Z}}}{\mathcal M}\), and \(| {\mathcal K}| <\mu\), where \(\mu\) is the least measurable cardinal, then \(\bigwedge {\mathcal K}>_{{\mathbb{Z}}}{\mathcal M}.\)
Theorem 3. If \({\mathcal W}>_{{\mathbb{Z}}}{\mathcal M}\) and \({\mathcal W}\) has fewer than \(\mu\) operations, then there exists a ring R with \(| R| <\mu\) such that \(_{{\mathbb{Z}}}{\mathcal M}<_ R{\mathcal M}\) and \({\mathcal W}\nleq_ R{\mathcal M}.\)
In some of the sections, some of the theorems are proved in a more general way. In section 9, Theorems 2 and 3 are proved with \(_{{\mathbb{Z}}}{\mathcal M}\) replaced by the variety of groups.

MSC:

08B15 Lattices of varieties
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