Ideal Extension

The extension of , an ideal in commutative ring , in a ring , is the ideal generated by its image under a ring homomorphism . Explicitly, it is any finite sum of the form where is in and is in . Sometimes the extension of is denoted .

The image may not be an ideal if is not surjective. For instance, 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.

The extension of a prime ideal may not be prime. For example, consider . Then the extension of the even integers is not a prime ideal since .