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

ABSTRACT

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.

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

Cite this paper

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.

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

References

[1] R G. Swan. “An Application of Graph Theory to Algebra”, Proc. Amer. Math. Soc. Vol.14,1963, pp.367-373. Correction, Proc. Amer. Soc. Vol.21,1969, pp.379-380.

[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.

[1] R G. Swan. “An Application of Graph Theory to Algebra”, Proc. Amer. Math. Soc. Vol.14,1963, pp.367-373. Correction, Proc. Amer. Soc. Vol.21,1969, pp.379-380.

[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.