# The rings which are Boolean

• Volume: 31, Issue: 2, page 175-184
• ISSN: 1509-9415

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## Abstract

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We study unitary rings of characteristic 2 satisfying identity ${x}^{p}=x$ for some natural number p. We characterize several infinite families of these rings which are Boolean, i.e., every element is idempotent. For example, it is in the case if $p={2}^{n}-2$ or $p={2}^{n}-5$ or $p={2}^{n}+1$ for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form ${2}^{q}+2m+1$ or ${2}^{q}+2m$ where q is a natural number and $m\in 1,2,...,{2}^{q}-1$.

## How to cite

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Ivan Chajda, and Filip Švrček. "The rings which are Boolean." Discussiones Mathematicae - General Algebra and Applications 31.2 (2011): 175-184. <http://eudml.org/doc/276738>.

@article{IvanChajda2011,
abstract = {We study unitary rings of characteristic 2 satisfying identity $x^p = x$ for some natural number p. We characterize several infinite families of these rings which are Boolean, i.e., every element is idempotent. For example, it is in the case if $p = 2^\{n\} - 2$ or $p = 2^\{n\} - 5$ or $p = 2^\{n\} + 1$ for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form $2^\{q\} + 2m + 1$ or $2^\{q\} + 2m$ where q is a natural number and $m ∈ \{1,2,...,2^\{q\} - 1\}$.},
author = {Ivan Chajda, Filip Švrček},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {Boolean ring; unitary ring; characteristic 2},
language = {eng},
number = {2},
pages = {175-184},
title = {The rings which are Boolean},
url = {http://eudml.org/doc/276738},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Ivan Chajda
AU - Filip Švrček
TI - The rings which are Boolean
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2011
VL - 31
IS - 2
SP - 175
EP - 184
AB - We study unitary rings of characteristic 2 satisfying identity $x^p = x$ for some natural number p. We characterize several infinite families of these rings which are Boolean, i.e., every element is idempotent. For example, it is in the case if $p = 2^{n} - 2$ or $p = 2^{n} - 5$ or $p = 2^{n} + 1$ for a suitable natural number n. Some other (more general) cases are solved for p expressed in the form $2^{q} + 2m + 1$ or $2^{q} + 2m$ where q is a natural number and $m ∈ {1,2,...,2^{q} - 1}$.
LA - eng
KW - Boolean ring; unitary ring; characteristic 2
UR - http://eudml.org/doc/276738
ER -

## References

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1. [1] I.T. Adamson, Rings, modules and algebras (Oliver&Boyd, Edinburgh, 1971) Zbl0226.16003
2. [2] G. Birkhoff, Lattice Theory, 3rd edition (AMS Colloq. Publ. 25, Providence, RI, 1979)
3. [3] I. Chajda and F. Švrček, Lattice-like structures derived from rings, Contributions to General Algebra, Proc. of Salzburg Conference (AAA81), J. Hayn, Klagenfurt 20 (2011), 11-18. Zbl1321.06011
4. [4] N. Jacobson, Structure of Rings (Amer. Math. Soc., Colloq. Publ. 36 (rev. ed.), Providence, RI, 1964).
5. [5] J. Lambek, Lectures on Rings and Modules (Blaisdell Publ. Comp., Waltham, Massachusetts, Toronto, London, 1966).
6. [6] N.H. McCoy, Theory of Rings (Mainillan Comp., New York, 1964).

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