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.
