# First Isomorphism Theorem/Rings

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## Theorem

Let $\phi: R \to S$ be a ring homomorphism.

Let $\map \ker \phi$ be the kernel of $\phi$.

Then:

- $\Img \phi \cong R / \map \ker \phi$

where $\cong$ denotes ring isomorphism.

## Proof

From Ring Homomorphism whose Kernel contains Ideal, let $J = \map \ker \phi$.

This gives the ring homomorphism $\mu: R / \map \ker \phi \to S$ as follows:

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That is:

- $\phi = \mu \circ \nu$

Then we have:

- $\map \ker \mu = \map \ker \phi / \map \ker \phi$

This is the null subring of $R / \map \ker \phi$ by Quotient Ring Defined by Ring Itself is Null Ring.

Then from Kernel is Trivial iff Monomorphism it follows that $\mu$ is a monomorphism.

From $\phi = \mu \circ \nu$, we have:

- $\Img \mu = \Img \phi$

It follows that $\mu$ is an isomorphism.

$\blacksquare$

## Also known as

This result is also referred to as the **First Fundamental Theorem on Ring Homomorphisms**.

## Also see

## Sources

- 1969: C.R.J. Clapham:
*Introduction to Abstract Algebra*... (previous) ... (next): Chapter $5$: Rings: $\S 24$. Homomorphisms: Theorem $47$ - 1970: B. Hartley and T.O. Hawkes:
*Rings, Modules and Linear Algebra*... (previous) ... (next): $\S 2.2$: Homomorphisms: Theorem $2.9$ - 1978: Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*... (previous) ... (next): $\S 60.3$ Factor rings