  The purpose of this paper is to investigate a relationship between the index
of reducibility and the Chern coefficient for primary ideals. Therefore, the
main result of this paper gives a characterization of a Cohen-Macaulay ring in
terms of its the index of reducibility, its Cohen-Macaulay type, and the Chern
coefficient for parameter ideals. As corollaries to the main theorem we
obtained the characterizations of a Gorenstein ring in term of its Chern
coefficient for parameter ideals.
