# ideals to and from matrices

## forming an ideal from a matrix

After defining a matrix (see inputting a matrix) , M, the ideal generated by the entries of the matrix is obtained using the command ideal.
 i1 : R = ZZ/101[a..e]; i2 : M = matrix{{a^2*b-c^2, a*b^2-d^3, c^5-d},{a^2*b, b*c*d, c^5}} o2 = | a2b-c2 ab2-d3 c5-d | | a2b bcd c5 | 2 3 o2 : Matrix R <--- R i3 : ideal M 2 2 2 2 3 5 5 o3 = ideal (a b - c , a b, a*b - d , b*c*d, c - d, c ) o3 : Ideal of R

## forming a matrix from an ideal

In much the same way we can construct a 1 by n (where n is the number of generators of I), matrix from the n generators of an ideal I using the command, generators.
 i4 : I = ideal(a^2*b-c^2+c*d, a*b^2-b*d^3, c^5,d+e); o4 : Ideal of R i5 : generators I o5 = | a2b-c2+cd -bd3+ab2 c5 d+e | 1 4 o5 : Matrix R <--- R
The abbreviation gens can be used for generators.