Depth And Regularity Of Monomial Ideals

Our paper is divided into four chapters.In the first and second chapters,we give an outline and some backgrounds of this paper.Let R= k[x1,x2,…,xn],T ?k[y1,y2,…,ym]be two polynomial rings over a field k,and S = R(?)kT = k[x1,…,xn,y1…,ym].Let I(?)R and J C T be two nonzero proper ideals.We give some properties of(I + J)s about the associated prime ideal and depth in terms of those of I and J in Chapter 3.Let I,J be two monomial ideals of Borel type in R and Q an arbitrary monomial ideal in S(J,Q are not necessarily proper ideals of the polynomial ring R).We prove in the fourth chapter that(IJ:Q)is also of Borel type.Moreover,reg(IJ:Q)<reg(I)+reg(J).In particular,reg(IJ)<reg(I)+ reg(J)and reg(Tm)? mreg(I).As a corollary,let I be a monomial ideal of Borel type and K a monomial complete intersection.Then reg(IK)<reg(I)+reg(K).