MTH 421 Theorems

Integral Domain Cancellation

Let a, b, c belong to an integral domain. If a ≠ 0 and ab = ac, then b = c

Finite Integral Domains

A finite integral domain is a field

Characteristic of a Ring with Unity

Let R be a ring with multiplicative identity. If 1 has infinite order under addition, then the characteristic of R is 0. If 1 has order n under addition, then the characteristic of R is n.

Characteristic of an Integral Domain

The characteristic of an integral domain is 0 or prime

R/A Is an Integral Domain If and Only If _______

Let R be a commutative ring with multiplicative identity and let A be an ideal of R. Then R/A is an integral domain if and only if A is prime.

R/A Is a Field If and Only If ______

Let R be a commutative ring with multiplicative identity and let A be an ideal of R. Then R/A is a field if and only if A is maximal

Kernels are _______

Ideals

Ideals are ______

Kernels

First Isomorphism Theorem for Rings

R/Ker 𝜙 is isomorphic to 𝜙(R).