On pseudocomplements and supplements in the big lattice of preradicals. (Q2874707)
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scientific article; zbMATH DE number 6327984
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | On pseudocomplements and supplements in the big lattice of preradicals. |
scientific article; zbMATH DE number 6327984 |
Statements
8 August 2014
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big lattices of preradicals
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pseudocomplements
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supplements
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essential preradicals
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superfluous preradicals
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0.8895627
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0.8850398
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0.88282126
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0.8819243
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0.8729532
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0.87154925
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On pseudocomplements and supplements in the big lattice of preradicals. (English)
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Let \(R\)-pr be the class of all preradicals of \(R\)-Mod, \(R\) an associative ring with identity. With respect to the usual ordering, \(R\)-pr is known to be a complete, atomic, coatomic, modular and upper continuous strongly pseudocomplemented big lattice. The authors continue investigations of this lattice. They describe all the nonzero preradicals that are essential and superfluous, respectively. Two alternative proofs that \(R\)-pr is a strongly pseudocomplemented big lattice are given. One of these is obtained by showing that \(R\)-pr is a strongly pseudocomplemented big lattice if and only if the class of all fully invariant submodules of an injective cogenerator \(E\) for \( R\)-Mod is a strongly pseudocomplemented lattice. A necessary and sufficient condition for \(R\)-pr to be a (strongly) supplemented big lattice is also given.
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