Examples of rings

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

Example 1

The integers, [ilmath]\mathbb{Z} [/ilmath] 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, [ilmath]\mathbb{R} [/ilmath] is a commutative ring with unity too, and [ilmath]\mathbb{Z} [/ilmath] is a subring of [ilmath]\mathbb{R} [/ilmath]

Example 1.2

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

Example 2

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

• This is a subring of [ilmath]\mathbb{R} [/ilmath], 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 [ilmath]\mathbb{R} [/ilmath], as are commutativity of addition and multiplication. That is:

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

Further examples

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