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
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {IV}$: Rings and Fields: $22$. New Rings from Old: Theorem $22.6: \ 2^\circ$