Difference between revisions of "Ring"
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Theorem: The additive identity of a ring {{M|R}} is unique (and as such can be denoted {{M|0}} unambiguously) | Theorem: The additive identity of a ring {{M|R}} is unique (and as such can be denoted {{M|0}} unambiguously) | ||
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* [[Examples of rings]] | * [[Examples of rings]] | ||
Revision as of 11:21, 21 May 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[1]:
| 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] |
Is a ring, which we write: (R,+:R×R→R,×:R×R→R) but because Mathematicians are lazy we write simply:
- (R,+,×)
Subring
If (S,+,×) is a ring, and every element of S is also in R (for another ring (R,+,×)) and the operations of addition and multiplication on S are the same as those on R (when restricted to S of course) then we say "S is a subring of R"
Note:
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
| Name | Statement | Explanation |
|---|---|---|
| Commutative Ring | ∀x,y∈R[xy=yx] | The order we multiply by does not matter. Calling a ring commutative isn't ambiguous because by definition addition in a ring is commutative so when we call a ring commutative we must mean "it is a ring, and also multiplication is commutative". |
| Ring with Unity | ∃e×∈R∀x∈R[xe×=e×x=x] | The existence of a multiplicative identity, once we have proved it is unique we often denote this "1" |
Using properties
A commutative ring with unity is a ring with the additional properties of:
- ∀x,y∈R[xy=yx]
- ∃e×∈R∀x∈R[xe×=e×x=x]
It is that simple.
Important theorem
a0=0a=0
use a(a+0)=aa and go from there.
Important theorems
Theorem: The additive identity of a ring R is unique (and as such can be denoted 0 unambiguously)
Theorem: if a+c=b+c then a=b (and due to commutivity of addition c+a=c+b⟹a=b too)
Theorem: The additive inverse of an element is unique (and herein, for a given x∈R shall be denoted −x)
See next
See also
References
- Jump up ↑ Fundamentals of abstract algebra - an expanded version - Neal H. McCoy
