Definition:Ring Homomorphism
Definition
Let $\struct {R, +, \circ}$ and $\struct {S, \oplus, *}$ be rings.
Let $\phi: R \to S$ be a mapping such that both $+$ and $\circ$ have the morphism property under $\phi$.
That is, $\forall a, b \in R$:
\(\text {(1)}: \quad\) | \(\ds \map \phi {a + b}\) | \(=\) | \(\ds \map \phi a \oplus \map \phi b\) | |||||||||||
\(\text {(2)}: \quad\) | \(\ds \map \phi {a \circ b}\) | \(=\) | \(\ds \map \phi a * \map \phi b\) |
Then $\phi: \struct {R, +, \circ} \to \struct {S, \oplus, *}$ is a ring homomorphism.
Also defined as
Sources dealing only with rings with unity may define a ring homomorphism as a unital ring homomorphism.
Also known as
A ring homomorphism is also known as a (ring) representation, but that is reserved on $\mathsf{Pr} \infty \mathsf{fWiki}$ for a more specialised definition.
Some sources hyphenate: ring-homomorphism.
Also see
- Definition:Ring Epimorphism: a surjective ring homomorphism
- Definition:Ring Monomorphism: an injective ring homomorphism
- Definition:Ring Isomorphism: a bijective ring homomorphism
- Definition:Ring Endomorphism: a ring homomorphism from a ring to itself
- Definition:Ring Automorphism: a ring isomorphism from a ring to itself
- Results about ring homomorphisms can be found here.
Linguistic Note
The word homomorphism comes from the Greek morphe (μορφή) meaning form or structure, with the prefix homo- meaning similar.
Thus homomorphism means similar structure.
Sources
- 1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Algebraic Concepts
- 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): Chapter $\text {I}$: Elementary Definitions: $\S 3$. Homomorphisms
- 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $5$: Rings: $\S 24$. Homomorphisms
- 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): $\S 2.2$: Homomorphisms: Definition $2.4$
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 7$: Homomorphisms and quotient algebras
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 57$. Ring homomorphisms
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): homomorphism
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): homomorphism