Examples of rings

Rings here refers to rings from abstract algebra, not a ring of sets

Example 1

The integers, $\mathbb{Z}$ with the usual operations of addition and multiplication (usual meaning 5+4 = 9, 5*4=20 - so forth) is a:

• Commutative ring with unity

Example 1.1

The real numbers, $\mathbb{R}$ is a commutative ring with unity too, and $\mathbb{Z}$ is a subring of $\mathbb{R}$

Example 1.2

The complex numbers $\mathbb{C}$ is a commutative ring with unity. $\mathbb{R}$ is a subring, and so is $\mathbb{Z}$

Example 2

Let

$$S=\{x+y\sqrt{2}\in\mathbb{R}|x,y\in\mathbb{Z}\}$$ , defining multiplication and addition in the usual way, this is a ring, infact:

• This is a subring of $\mathbb{R}$, it is a commutative ring with unity.

Proof

To prove it is a ring one must verify the "axioms of a ring" (found on the ring page at the top), but to sum up one must show:

• Multiplication is closed
• Addition is closed
• The identities (both additive and multiplicative) are in the ring
• The additive inverse is in the ring
• Multiplication is commutative

Associativity of addition and multiplication are "inherited" from $\mathbb{R}$, as are commutativity of addition and multiplication. That is:

• As $x=(a+b\sqrt{2})\in\mathbb{R}$ and $y=(c+d\sqrt{2})\in\mathbb{R}$ we know automatically as multiplication is commutative in $\mathbb{R}$ for example.

Further examples

• Table on p25 of Fundamentals of Abstract Algebra, Neal H. McCoy, an expanded version might be good.