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On Boolean subrings of rings. (English) Zbl 1332.16018
Fontana, Marco (ed.) et al., Commutative algebra. Recent advances in commutative rings, integer-valued polynomials, and polynomial functions. Based on mini-courses and a conference on commutative rings, integer-valued polynomials and polynomial functions, Graz, Austria, December 16–18 and December 19–22, 2012. New York, NY: Springer (ISBN 978-1-4939-0924-7/hbk; 978-1-4939-0925-4/ebook). 113-117 (2014).
In the paper the authors study the maximal subrings that satisfy the identity \(x^{p+k}=x^p\), for some \(p,k\geq 1\). An explicit description of the subrings is given in the case \(k=2^s\) or \(k=2^s-1\), for some \(s\geq 0\).
For the entire collection see [Zbl 1294.13002].
MSC:
16R40 Identities other than those of matrices over commutative rings
06E20 Ring-theoretic properties of Boolean algebras
13M05 Structure of finite commutative rings
16P10 Finite rings and finite-dimensional associative algebras
16U80 Generalizations of commutativity (associative rings and algebras)
16U70 Center, normalizer (invariant elements) (associative rings and algebras)
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References:
[1] G. Birkhoff, \(Lattice Theory\), 3rd edn. Colloquium Publications, vol. 25 (American Mathematical Society, Providence, 1967)
[2] I. Chajda, F. Švrček, Lattice-like structures derived from rings. In \(Contributions to General Algebra\), vol. 20 (Verlag Johannes Heyn, Klagenfurt, 2012), pp. 11-18 · Zbl 1321.06011
[3] I. Chajda, F. Švrček, The rings which are Boolean. Discuss. Mathem. General Algebra Appl. 31, 175-184 (2011) · Zbl 1262.06005 · doi:10.7151/dmgaa.1181
[4] P. Jedlička, The rings which are Boolean II. Acta Univ. Carolinae (to appear)
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