# The rings which are Boolean

Discussiones Mathematicae - General Algebra and Applications (2011)

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

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topIvan 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

top- [1] I.T. Adamson, Rings, modules and algebras (Oliver&Boyd, Edinburgh, 1971) Zbl0226.16003
- [2] G. Birkhoff, Lattice Theory, 3rd edition (AMS Colloq. Publ. 25, Providence, RI, 1979)
- [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] N. Jacobson, Structure of Rings (Amer. Math. Soc., Colloq. Publ. 36 (rev. ed.), Providence, RI, 1964).
- [5] J. Lambek, Lectures on Rings and Modules (Blaisdell Publ. Comp., Waltham, Massachusetts, Toronto, London, 1966).
- [6] N.H. McCoy, Theory of Rings (Mainillan Comp., New York, 1964).

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