Classification of rings with toroidal Jacobson graph

Selvakumar, Krishnan; Subajini, Manoharan

doi:10.1007/s10587-016-0257-y

Geodesic

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Classification of rings with toroidal Jacobson graph

Selvakumar, Krishnan ; Subajini, Manoharan

Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 307-316.

Voir la notice de l'article dans Czech Digital Mathematics Library

Résumé

Let

R

be a commutative ring with nonzero identity and

J (R)

the Jacobson radical of

R

. The Jacobson graph of

R

, denoted by

J_{R}

, is defined as the graph with vertex set

R ∖ J (R)

such that two distinct vertices

x

and

y

are adjacent if and only if

1 - x y

is not a unit of

R

. The genus of a simple graph

G

is the smallest nonnegative integer

n

such that

G

can be embedded into an orientable surface

S_{n}

. In this paper, we investigate the genus number of the compact Riemann surface in which

J_{R}

can be embedded and explicitly determine all finite commutative rings

R

(up to isomorphism) such that

J_{R}

is toroidal.

MR Zbl

DOI : 10.1007/s10587-016-0257-y

Classification : 05C10, 05C25, 13M05
Mots-clés : planar graph; genus of a graph; local ring; nilpotent element; Jacobson graph

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     title = {Classification of rings with toroidal {Jacobson} graph},
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     publisher = {mathdoc},
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Selvakumar, Krishnan; Subajini, Manoharan. Classification of rings with toroidal Jacobson graph. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 307-316. doi : 10.1007/s10587-016-0257-y. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-016-0257-y/

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