Preimage of Image of Ideal under Ring Homomorphism

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Theorem

Let $\phi: \struct {R_1, +_1, \circ_1} \to \struct {R_2, +_2, \circ_2}$ be a ring homomorphism.

Let $K = \map \ker \phi$, be the kernel of $\phi$.

Let $J$ be an ideal of $R_1$.


Then:

$\phi^{-1} \sqbrk {\phi \sqbrk J} = J + K$


Proof

As an ideal is a subring, the result Preimage of Image of Subring under Ring Homomorphism applies directly.

$\blacksquare$


Sources