Abstract Algebra/The hierarchy of rings

Commutative rings

Definition 11.1:

A ring $R$ with multiplication $\cdot$ is called commutative if and only if $a\cdot b=b\cdot a$ for all $a,b\in R$ .

Examples 11.2:

The whole numbers $\mathbb {Z}$ are commutative.
The matrix ring Failed to parse (unknown function "\middle"): {\displaystyle \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \middle| a, b, c, d \in \mathbb R, ad - bc \neq 0 \right\}} with matrix multiplication and component-wise addition is not commutative.

In commutative rings, a left ideal is a right ideal and thus a two-sided ideal, and a right ideal also.

Integral domains

Definition 11.3:

An integral domain is defined to be a commutative ring (that is, we assume commutativity by definition) such that whenever $ab=0$ ( $a,b\in R$ ), then $a=0$ or $b=0$ .

We can characterize integral domains in another way, and this involves the so-called zero-divisors.

Definition 11.4:

Let $R$

Thus, a ring is an integral domain iff it has no zero divisors.

Principal ideal domains

Theorem 11.?:

Every PID is a UFD.

Proof:

Let $R$ be a PID, and let $a\in R$ .

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