Special Types of Rings Review

# Special Types of Rings Review

We will now review some of the recent material regarding various special types of rings. We summarize this information in the table below.

Type of Ring Special Properties
Commutative Rings A ring $(R, +, *)$ is a commutative ring if for all $a, b \in R$ we have that $a * b = b * a$. In other words, a commutative ring is a ring that is multiplicatively commutative.
Division Rings A ring $(R, +, *)$ is a division ring (with multiplicative identity $1$) if for all $x \in R$ with $x \neq 0$ there exists an element $x^{-1} \in R$ such that $x * x^{-1} = 1$ and $x^{-1} * x = 1$. In other words, division rings are rings such that every nonzero element in $R$ has a multiplicative inverse in $R$.
Integral Domains
- Zero Divisors in Rings
- Zero Divisors in Z/nZ
- The Cancellation Property of * on Integral Domains
- The Integral Domain of Z/pZ
A ring $(R, +, *)$ is an integral domain if it is a commutative ring AND has no zero divisors, i.e., for all $a, b \in R$ if $a * b = 0$ then $a = 0$ or $b = 0$ (or both). In other words, integral domains are commutative rings such that multiplication by two nonzero elements in $R$ yields a nonzero element in $R$.