# Ring elements as sums of units

Open Mathematics (2009)

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

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topCharles 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

top- [1] Jacobson N., Structure of rings, American Mathematical Society Colloquium Publications, 1956, 37 Zbl0073.02002
- [2] Lovász L., Combinatorial problems and exercises, North-Holland, Amsterdam, 1979 Zbl0439.05001
- [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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