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  text: '[summary: In [3gq ring theory], the notion of "[ideal_ring_theory ideal]" corresponds precisely with the notion of "[5r6 kernel] of [-ring_homomorphism]".]\n\nIn [3gq ring theory], the notion of "[ideal_ring_theory ideal]" corresponds precisely with the notion of "[5r6 kernel] of [-ring_homomorphism]".\n\nThis result is analogous to the fact from [3gd group theory] that [4h6 normal subgroups] are the same thing as [49y kernels of group homomorphisms] ([4h7 proof]).\n\n# Proof\n\n## Kernels are ideals\n\nLet $f: R \\to S$ be a ring homomorphism between rings $R$ and $S$.\nWe claim that the kernel $K$ of $f$ is an ideal.\n\nIndeed, it is clearly a [-576] of the ring $R$ when viewed as just an additive group %%note:That is, after removing the multiplicative structure from the ring.%% because $f$ is a *group* homomorphism between the underlying additive groups, and kernels of *group* homomorphisms are subgroups (indeed, *normal* subgroups). ([4h7 Proof.])\n\nWe just need to show, then, that $K$ is closed under multiplication by elements of the ring $R$.\nBut this is easy: if $k \\in K$ and $r \\in R$, then $f(kr) = f(k)f(r) = 0 \\times r = 0$, so $kr$ is in $K$ if $k$ is.\n\n## Ideals are kernels\n\n[todo: refer to the quotient group, and therefore introduce the quotient ring]',
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