The extension of a, an ideal in commutative ring A, in a ring B, is the ideal generated by its image f(a) under a ring homomorphism f. Explicitly, it is any finite sum of the form sum y_i f(x_i) where y_i is in B and x_i is in a. Sometimes the extension of a is denoted a^e. The image f(a) may not be an ideal if f is not surjective. For instance, f:Z->Z[x] is a ring homomorphism and the image of the even integers is not an ideal since it does not contain any nonconstant polynomials. The extension of the even integers in this case is the set of polynomials with even coefficients.

algebraic number theory | ideal | ideal contraction | prime ideal | ring