Definition:Ring Isomorphism

Definition

Let $\left({R, +, \circ}\right)$ and $\left({S, \oplus, *}\right)$ be rings.

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

Then $\phi$ is a ring isomorphism iff $\phi$ is a bijection.

That is, $\phi$ is a ring isomorphism iff $\phi$ is both a monomorphism and an epimorphism.

Sources

• Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.3$
• Seth Warner: Modern Algebra (1965): $\S 23$
• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 5.24$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 2.2$: Definition $2.4$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 57$ Remarks: $\text{(a) (3), (b)}$