Cohen-Macaulayness and sequentially Cohen-Macaulayness of monomial ideals

In this paper, we give a characterization for Cohen-Macaulay rings R/I where I subset of R = K[y(1 ),..., y(n) ] is a monomial ideal which satisfies bigsize I = size I. Next, we let S = K[x(1) ,..., x(m),y(1 ),..., y(n)] be a polynomial ring and I subset of S a monomial ideal. We study the sequentially Cohen-Macaulayness of S / I with respect to Q = (y(1 ),..., y(n)). Moreover, if I subset of R is a monomial ideal such that the associated prime ideals of I are in pairwise disjoint sets of variables, a classification of R/I to be sequentially Cohen-Macaulay is given. Finally, we compute grade(Q, M) where M is a sequentially Cohen-Macaulay S-module with respect to Q.