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**An Application of Eulerian Graph to PI on Mn(C)** ()

We obtain a new class of polynomial identities on the ring of n

*×*n matrices over any commutative ring with 1 by using the Swan’s graph theoretic method [1] in the proof of Amitsur-Levitzki theorem. Let be an Eulerian graph with k vertices and d edges. Further let be an integer and assume that . We prore that is an PI on*Mn*(*C*). Standard and Chang [2] -Giambruno-Sehgal [3] polynomial identities are the spectial examples of our conclusions.Share and Cite:

S. You, H. Zhao, Y. Feng and M. Cao, "An Application of Eulerian Graph to PI on

*Mn*(*C*),"*Applied Mathematics*, Vol. 3 No. 7, 2012, pp. 809-811. doi: 10.4236/am.2012.37121.Conflicts of Interest

The authors declare no conflicts of interest.

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[2] | Q. Chang. “Some Consequences of the Standard Polynomial”. Proc. Amer. Math. Soc. Vol.104,1988, pp.707-710. |

[3] | A. Giambruno. S K. Sehgal. “On a Polynomial Idntity for n?n Matrices”. J.Algebra, Vol.126,1989, pp.451-453. |

[4] | S.F.You, Y.M.Zheng and D.G.Hu.“Eulerian Graph and Polynomial Identities on Matrix Rings”. Advances in Math. Vol.32,2003, pp.425-428 |

[5] | S.F.You. “The Primitivity of Extended Centroid Extension on Prime GPI-rings”. Advances in Math. Vol.29,2000, pp.331-336. |

[6] | S.F.You. “The Essential (one-sided) Ideal of Semiprime PI-Rings”. Acta. Math. Sinica. Vol.44,2001, pp.747-752. |

[7] | S.F.You, M.Cao and Y.J.Feng, “Semiautomata and Near Rings”, Quantitative Logic and Soft Computing 5, World Scientific, 2012, pp.428-431. |

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