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Annihilator characterizations of Boolean rings and Boolean lattices. (English. Russian original) Zbl 0805.06013

Math. Notes 53, No. 2, 124-129 (1993); translation from Mat. Zametki 53, No. 2, 15-24 (1993).
An associative ring is said to be Boolean provided all its elements are idempotent. A generalized Boolean lattice is defined as a relatively pseudocomplemented distributive lattice with zero. The annihilator of an element \(a\) of a ring \(R\) (lattice \(L\) with zero) is the set \(a^*= \{x\in R\mid ax= xa= 0\}\) (\(a^*= \{x\in L\mid a\land x= 0\}\)). Various characterizations of Boolean rings within the class of associative rings (of generalized Boolean lattices within the class of distributive lattices with zero) are provided in terms of annihilators. Boolean rings are also characterized in terms of 0-automorphisms, i.e. of bijections \(\alpha\) such that \(xy= 0\Leftrightarrow \alpha(x)\alpha(y)= 0\).

MSC:

06E20 Ring-theoretic properties of Boolean algebras
16B99 General and miscellaneous
06D15 Pseudocomplemented lattices
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References:

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