Introduction to Ideals

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4.11.1 Introduction to Ideals

An object of type IDEAL in CoCoA represents an ideal. An ideal is formed using the command Ideal(P_1,...,P_n) where the P_i are generators for the ideal.

Example

  Use R ::= QQ[x,y];
  I := Ideal(x,y^2,2+xy^2);

The following algebraic operations yield ideals:

  I^N, I+J, E*F, P:Q

where: I and J are ideals; N is a non-negative integer; E, F are either both ideals or one is an ideal and the other is a polynomial; and the pair (P, Q) has the form (IDEAL, POLY), (IDEAL, IDEAL), (MODULE, VECTOR), (MODULE, MODULE).

Example

  Use R ::= QQ[x,y];
  I := Ideal(x,y^2,2+xy^2);
  I^2;
Ideal(x^2, xy^2, x^2y^2 + 2x, y^4, xy^4 + 2y^2, x^2y^4 + 4xy^2 + 4)
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  J := Ideal(y);
  I+J;
Ideal(x, y^2, xy^2 + 2, y)
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  I*J;
Ideal(xy, y^3, xy^3 + 2y)
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