Extension and contraction of ideals

A ring homomorphism transports ideals forward by extension and backward by inverse image, or contraction.

Table of contents
- Extension of an ideal Extension and contraction of ideals
- Contraction of an ideal Extension and contraction of ideals

Extension of an ideal ( $I^{e}$ )

For a ring homomorphism

f : R \to A

and an ideal

I \subseteq R

, the extension

I^{e} = I A

is the ideal of

A

generated by

f (I)

.

Contraction of an ideal ( $J^{c}$ )

For a ring homomorphism

f : R \to A

and an ideal

J \subseteq A

, its contraction is

J^{c} = f^{- 1} (J)

.

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