Ideal Extension

The ideal extension of , an ideal in a commutative ring , in a commutative ring with a multiplicative identity, is the ideal generated by its image under a ring homomorphism . Explicitly, it consists of all finite sums 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 a surjection. 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 a prime ideal. For example, consider . Then the extension of the even integers is not a prime ideal since .