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Algebraic representation of mappings between submodule lattices

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Abstract

We show that under certain weak conditions on the module R M, every mapping

$$ f:\mathfrak{L}\left( {_R M} \right) \to \mathfrak{L}\left( {_S N} \right) $$

between the submodule lattices which preserves arbitrary joins and “disjointness” has a unique representation of the form f(u) = 〈h[ S B R × R U]〉 for all u ∈

$$ \mathfrak{L}\left( {_R M} \right) $$

, where S B R is some bimodule and h is an R-balanced mapping. Furthermore, f is a lattice homomorphism if and only if B R is flat and the induced S-module homomorphism

$$ \bar h:_S B \otimes _R M \to _S N $$

is monic. If S N also satisfies the same weak conditions, then f is a lattice isomorphism if and only if B R is a finitely generated projective generator, S ≅ End(B R ) canonically, and

$$ \bar h:_S B \otimes _R M \to _S N $$

is an S-module isomorphism, i.e., every lattice isomorphism is induced by a Morita equivalence between R and S and a module isomorphism.

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  1. Fakultät Mathematik und Naturwissenschaften, Fachrichtung Mathematik, Hausanschrift: Willersbau, Zellescher Weg 12-14, 01069, Dresden, Germany

    Ulrich Brehm

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Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 46, Algebra, 2007.

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Brehm, U. Algebraic representation of mappings between submodule lattices. J Math Sci 153, 454–480 (2008). https://doi.org/10.1007/s10958-008-9131-2

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