First Isomorphism Theorem/Rings
Let $\phi: R \to S$ be a ring homomorphism.
Let $\map \ker \phi$ be the kernel of $\phi$.
- $\Img \phi \cong R / \map \ker \phi$
where $\cong$ denotes ring isomorphism.
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:
- $\phi = \mu \circ \nu$
Then we have:
- $\map \ker \mu = \map \ker \phi / \map \ker \phi$
From $\phi = \mu \circ \nu$, we have:
- $\Img \mu = \Img \phi$
It follows that $\mu$ is an isomorphism.
Also known as
This result is also referred to as the First Fundamental Theorem on Ring Homomorphisms.
- 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