Extension and contraction of ideals

In commutative algebra, the extension and contraction of ideals are operations performed on sets of ideals.

Let A and B be two commutative rings with unity, and let f : A → B be a (unital) ring homomorphism. If ${\displaystyle {\mathfrak {a}}}$ is an ideal in A, then ${\displaystyle f({\mathfrak {a}})}$ need not be an ideal in B (e.g. take f to be the inclusion of the ring of integers Z into the field of rationals Q). The extension ${\displaystyle {\mathfrak {a}}^{e}}$ of ${\displaystyle {\mathfrak {a}}}$ in B is defined to be the ideal in B generated by ${\displaystyle f({\mathfrak {a}})}$. Explicitly,

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