Difference between revisions of "Examples of rings"
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Associativity of addition and multiplication are "inherited" from {{M|\mathbb{R} }}, as are commutativity of addition and multiplication. That is: | Associativity of addition and multiplication are "inherited" from {{M|\mathbb{R} }}, as are commutativity of addition and multiplication. That is: | ||
* As {{M|1=x=(a+b\sqrt{2})\in\mathbb{R} }} and {{M|1=y=(c+d\sqrt{2})\in\mathbb{R} }} we know automatically {{M|xy=yx}} as multiplication is commutative in {{M|\mathbb{R} }} for example. | * As {{M|1=x=(a+b\sqrt{2})\in\mathbb{R} }} and {{M|1=y=(c+d\sqrt{2})\in\mathbb{R} }} we know automatically {{M|xy=yx}} as multiplication is commutative in {{M|\mathbb{R} }} for example. | ||
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| + | ==Further examples== | ||
| + | * Table on p25 of Fundamentals of Abstract Algebra, Neal H. McCoy, an expanded version might be good. | ||
{{Example|Abstract Algebra}} | {{Example|Abstract Algebra}} | ||
Latest revision as of 17:11, 19 May 2015
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.
