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# 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.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Macaulay2Doc/ov_ideals.m2:107:0.