Ring elements as sums of units

Charles Lanski; Attila Maróti

Open Mathematics (2009)

  • Volume: 7, Issue: 3, page 395-399
  • ISSN: 2391-5455

Abstract

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In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.

How to cite

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Charles Lanski, and Attila Maróti. "Ring elements as sums of units." Open Mathematics 7.3 (2009): 395-399. <http://eudml.org/doc/269249>.

@article{CharlesLanski2009,
abstract = {In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.},
author = {Charles Lanski, Attila Maróti},
journal = {Open Mathematics},
keywords = {Artinian rings; Units; Hamiltonian cycle; sums of units; Hamiltonian cycles; finite rings},
language = {eng},
number = {3},
pages = {395-399},
title = {Ring elements as sums of units},
url = {http://eudml.org/doc/269249},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Charles Lanski
AU - Attila Maróti
TI - Ring elements as sums of units
JO - Open Mathematics
PY - 2009
VL - 7
IS - 3
SP - 395
EP - 399
AB - In an Artinian ring R every element of R can be expressed as the sum of two units if and only if R/J(R) does not contain a summand isomorphic to the field with two elements. This result is used to describe those finite rings R for which Γ(R) contains a Hamiltonian cycle where Γ(R) is the (simple) graph defined on the elements of R with an edge between vertices r and s if and only if r - s is invertible. It is also shown that for an Artinian ring R the number of connected components of the graph Γ(R) is a power of 2.
LA - eng
KW - Artinian rings; Units; Hamiltonian cycle; sums of units; Hamiltonian cycles; finite rings
UR - http://eudml.org/doc/269249
ER -

References

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  1. [1] Jacobson N., Structure of rings, American Mathematical Society Colloquium Publications, 1956, 37 Zbl0073.02002
  2. [2] Lovász L., Combinatorial problems and exercises, North-Holland, Amsterdam, 1979 Zbl0439.05001
  3. [3] Lucchini A., Maróti A., Some results and questions related to the generating graph of a finite group, Proceedings of the Ischia Group Theory Conference 2008 (to appear) Zbl1191.20018

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