Journal of Mathematical Sciences

, 153:454 | Cite as

Algebraic representation of mappings between submodule lattices

  • Ulrich Brehm


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.


Projective Geometry Geometric Algebra Regular Ring Picard Group Lattice Homomorphism 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Fakultät Mathematik und NaturwissenschaftenFachrichtung Mathematik, Hausanschrift: WillersbauDresdenGermany

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