Abstract Algebra: A Brief Summary of Ring Theory

Introduction to rings

Definition: ring

A ring $R$ is a set together with two binary operations $+$ and $\times$ (called addition and multiplication) satisfying the following axioms:

1. $(R,+)$ is an abelian group,
2. $\times$ is associative: $(a\times b) \times c = a\times (b\times c)$, for all $a,b,c\in R$,
3. the distributive laws hold in $R$: for all $a,b,c\in R$

\[\begin{equation} (a+b)\times c = (a\times c) + (b\times c), \quad \text{and} \quad a\times (b+c) = (a\times b) + (a\times c) \end{equation}\]

• The ring is commutative if multiplication is commutative.
• The ring $R$ is said to have an identity (or contain a $1$) if there is an element $1\in R$ with

\[\begin{equation} 1\times a = a\times 1, \quad \text{for all $a\in R$}. \end{equation}\]

A ring $R$ with identity $1$, where $1 \neq 0$, is called a division ring (or skew field) if every nonzero element $a\in R$ has a multiplicative inverse, i.e., there exists $b \in R$ such that $ab=ba=1$. A commutative division ring is called a field.

Proposition

Let $R$ be a ring. Then

1. $0a=a0=0$ for all $a\in R$.
2. $(-a)b = a(-b) = -(ab)$ for all $a,b\in R$ (recall $-a$ is the additive inverse of $a$).
3. $(-a)(-b) = ab$ for all $a,b\in R$.
4. if $R$ has an identity $1$, then the identity is unique and $-a = (-1)a$.

Unique factorization domains

Chain of class inclusions:

rngs $\supset$ rings $\supset$ commutative rings $\supset$ integral domains $\supset$ integrally closed domains $\supset$ GCD domains $\supset$ unique factorization domains $\supset$ principal ideal domains $\supset$ Euclidean domains $\supset$ fields $\supset$ algebraically closed fields

Polynomial rings

to be continued …

References

1. Dummit, D. S., and Foote, R. M. Abstract Algebra, 3rd ed. John Wiley & Sons, Inc. (2004).
2. Artin, M. Algebra, 2nd ed. Pearson Education. (2011).
3. Judson, T. M. Abstract Algebra: Theory and Application. Orthogonal Publishing L3c. (2022).