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sums, products, and powers of ideals

Arithmetic for ideals uses the standard symbols. Below are examples of the basic arithmetic functions for ideal.
i1 : R = ZZ/101[a..d]/(b*c-a*d,c^2-b*d,b^2-a*c);
For more information about quotient rings see quotient rings.
i2 : I = ideal (a*b-c,d^3);

o2 : Ideal of R
i3 : J = ideal (a^3,b*c-d);

o3 : Ideal of R
i4 : I+J

                      3   3
o4 = ideal (a*b - c, d , a , a*d - d)

o4 : Ideal of R
i5 : I*J

             4     3    2                            3 3     4    4
o5 = ideal (a b - a c, a b*d - a*b*d - a*c*d + c*d, a d , a*d  - d )

o5 : Ideal of R
i6 : I^2

             3      2              3      3   6
o6 = ideal (a c - 2a d + b*d, a*b*d  - c*d , d )

o6 : Ideal of R
For more information see Ideal + Ideal, Ideal * Ideal, and Ideal ^ ZZ.