Difference between revisions of "Ring"
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Some books introduce rings first, I do not know why. A ring is an additive [[Group|group]] (it is commutative making it an Abelian one at that), that is a ring is just a group {{M|(G,+)}} with another operation on {{M|G}} called {{M|\times}} | Some books introduce rings first, I do not know why. A ring is an additive [[Group|group]] (it is commutative making it an Abelian one at that), that is a ring is just a group {{M|(G,+)}} with another operation on {{M|G}} called {{M|\times}} | ||
| + | |||
| + | ==Properties== | ||
| + | {{Todo|I did these in a rush - just here for basic ref}} | ||
| + | ===Commutative ring=== | ||
| + | Multiplication is commutative | ||
| + | |||
| + | ===Ring with unity=== | ||
| + | There is a multiplicative identity | ||
| + | |||
| + | ==Multiplicative inverse== | ||
| + | For a ring with unity, if there exists an element s, such that as=sa=e then we call that the multiplicative inverse | ||
| + | |||
| + | ==Important theorem== | ||
| + | a0=0a=0 | ||
| + | |||
| + | use a(a+0)=aa and go from there. | ||
| + | |||
{{Definition|Abstract Algebra}} | {{Definition|Abstract Algebra}} | ||
Revision as of 13:04, 18 March 2015
Not to be confused with rings of sets which are a topic of algebras of sets and thus σ-Algebras and σ-rings
Contents
[hide]Definition
A set R and two binary operations + and × such that the following hold:
| Rule | Formal | Explanation |
|---|---|---|
| Addition is commutative | ∀a,b∈R[a+b=b+a] |
It doesn't matter what order we add |
| Addition is associative | ∀a,b,c∈R[(a+b)+c=a+(b+c)] |
Now writing a+b+c isn't ambiguous |
| Additive identity | ∃e∈R∀x∈R[e+x=x+e=x] |
We do not prove it is unique (after which it is usually denoted 0), just "it exists" The "exists e forall x∈R" is important, there exists a single e that always works |
| Additive inverse | ∀x∈R∃y∈R[x+y=y+x=e] |
We do not prove it is unique (after we do it is usually denoted −x, just that it exists The "forall x∈R there exists" states that for a given x∈R a y exists. Not a y exists for all x |
| Multiplication is associative | ∀a,b,c∈R[(ab)c=a(bc)] |
|
| Multiplication is distributive | ∀a,b,c∈R[a(b+c)=ab+ac] ∀a,b,c∈R[(a+b)c=ac+bc] |
Some books introduce rings first, I do not know why. A ring is an additive group (it is commutative making it an Abelian one at that), that is a ring is just a group (G,+) with another operation on G called ×
Properties
TODO: I did these in a rush - just here for basic ref
Commutative ring
Multiplication is commutative
Ring with unity
There is a multiplicative identity
Multiplicative inverse
For a ring with unity, if there exists an element s, such that as=sa=e then we call that the multiplicative inverse
Important theorem
a0=0a=0
use a(a+0)=aa and go from there.
