//SINGULAR Example 1.8.2


ring A  = 0,(x,y),dp;       
ideal I = x10+x9y2,y8-x2y7;
poly f  = xy13+y12;
matrix M=lift(I,f);        //f=M[1,1]*I[1]+...+M[r,1]*I[r]
M;

f-M[1,1]*I[1]-M[2,1]*I[2]; //test

ring R = 0,(x,y,z),ds;
poly f = yx2+yx;
ideal I = x-x2,y+x;

list L = division(f,I);            //division with remainder
L;

matrix(I)*L[1] - matrix(f)*L[2] - matrix(L[3]);   //test