Ring
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Contents
Not to be confused with a ring of sets
Definition
Let [ilmath]R[/ilmath] be a non-empty set, let there be two binary operations (a kind of map where rather than [ilmath]f(a,b)[/ilmath] we write [ilmath]afb[/ilmath]):
- [ilmath]\oplus:R\times R\rightarrow R[/ilmath] - called "addition", [ilmath]\oplus:(a,b)\mapsto a\oplus b[/ilmath]
- [ilmath]\odot:R\times R\rightarrow R[/ilmath] - called "multiplication", [ilmath]\odot:(a,b)\mapsto a\odot b[/ilmath]
and let there be elements [ilmath]0_R\in R[/ilmath] and [ilmath]1_R\in R[/ilmath] (not necessarily distinct)^{[Note 1]} such that we have the following 7 properties^{[1]}:
TODO: This would be much nicer as a table....
- [ilmath](R,\oplus,0_R)[/ilmath] is an abelian group
- Group definition:
- [ilmath]\forall a,b,c\in R[(a\oplus b)\oplus c=a\oplus(b\oplus c)][/ilmath] - associativity
- [ilmath]\exists e\in R\ \forall a\in R[e\oplus a=a\oplus e=a][/ilmath] - existence of identity, on the group page we show it is unique^{[Note 2]}, we denote it by [ilmath]0_R[/ilmath], so: [ilmath]\forall a\in R[a\oplus 0_R=0_R\oplus a=a][/ilmath]
- [ilmath]\forall a\in R\ \exists b\in R[a\oplus b=b\oplus a=0_R][/ilmath] - existence of inverse, on the group page we show it is unique^{[Note 3]}. Denoted by [ilmath]-a[/ilmath] as we're using additive notation^{[Note 4]}
- Being an Abelian group adds an additional property:
- [ilmath]\forall a,b\in R[a\oplus b=b\oplus a][/ilmath] - commutivity
- Group definition:
- [ilmath](R,\odot)[/ilmath] is a semigroup
- Semigroup definition:
- [ilmath]\forall a,b,c\in R[(a\odot b)\odot c=a\odot(b\odot c)][/ilmath]
- Semigroup definition:
- There is distributivity in play in.
- [ilmath]\odot[/ilmath] distributes across [ilmath]\oplus[/ilmath] Caution:I think... it might be the other way around... the following 2 rules are certainly correct however:
- [ilmath]\forall a,b,c\in R[a\odot(b\oplus c)=(a\odot b)\oplus(a\odot c)][/ilmath] and
- [ilmath]\forall a,b,c\in R[(a+b)c=ac+bc][/ilmath]
- [ilmath]\odot[/ilmath] distributes across [ilmath]\oplus[/ilmath] Caution:I think... it might be the other way around... the following 2 rules are certainly correct however:
Then [ilmath](R,\oplus:R\times R\rightarrow R,\odot:R\times R\rightarrow R,0_R)[/ilmath] is a ring, but as mathematicians are lazy we just write [ilmath](R,\oplus,\odot,0_R)[/ilmath], [ilmath](R,\oplus,\odot)[/ilmath] or even just "Let [ilmath]R[/ilmath] be a ring".
TODO: Be more formal about distributivity, I've checked my books, no one specified, they just say "it is distributive: "
Further properties of elementary rings
There are 2 more additional properties we can apply to define rings:
- [ilmath]\exists e_\odot\ \forall a\in R[a\odot e_\odot=e_\odot\odot a=a][/ilmath] - a multiplicative identity, this element if it exists is unique and denoted [ilmath]1_R[/ilmath] or just [ilmath]1[/ilmath]
- [ilmath]\forall a,b\in R[a\odot b=b\odot a][/ilmath] - commutative with respect to [ilmath]\odot[/ilmath]
Giving us the following 4 types of elementary rings^{[Note 5]}:
- Ring - properties 1-7
- Ring with unity (AKA: u-ring, ring with identity) - properties 1-8
- Commutative ring (AKA: c-ring) - properties 1-7 and 9
- Commutative ring with unity (AKA: cu-ring or q-ring - properties 1-9
Caveats
Some authors define a ring to be what we would call a ring with unity (which we shall call a u-ring throughout the site). Especially if the book covers the topics of rings and modules. We defined "commutative ring" and "ring with unity" above.
See next
- Types of ring
- Ring morphism
- Ring homomorphism
- Kernel of a ring homomorphism - see also: kernel
- Image of a ring homomorphism - see also: image
- Ring isomorphism
- Ring homomorphism
- Unit of a ring
- Division ring
- Ring ideal
- Quotient ring
- Fundamental ring homomorphism theorem
- Ring isomorphism theorems
- Module
Notes
- ↑ So we could have [ilmath]0_R=1_R[/ilmath] or we could have [ilmath]0_R\ne 1_R[/ilmath]
- ↑ there is only one inverse
- ↑ there is only one inverse for an element
- ↑ For multiplicative notation we'd use [ilmath]a^{-1} [/ilmath]
- ↑ field, integral domain are also all rings, there's like 6 kinds. We call "Elementary ring" just the ones listed
References
OLD PAGE
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Not to be confused with rings of sets which are a topic of algebras of sets and thus [ilmath]\sigma[/ilmath]-Algebras and [ilmath]\sigma[/ilmath]-rings
Definition
A set [ilmath]R[/ilmath] and two binary operations [ilmath]+[/ilmath] and [ilmath]\times[/ilmath] such that the following hold^{[1]}:
Rule | Formal | Explanation |
---|---|---|
Addition is commutative | [math]\forall a,b\in R[a+b=b+a][/math] | It doesn't matter what order we add |
Addition is associative | [math]\forall a,b,c\in R[(a+b)+c=a+(b+c)][/math] | Now writing [ilmath]a+b+c[/ilmath] isn't ambiguous |
Additive identity | [math]\exists e\in R\forall x\in R[e+x=x+e=x][/math] | We do not prove it is unique (after which it is usually denoted 0), just "it exists" The "exists [ilmath]e[/ilmath] forall [ilmath]x\in R[/ilmath]" is important, there exists a single [ilmath]e[/ilmath] that always works |
Additive inverse | [math]\forall x\in R\exists y\in R[x+y=y+x=e][/math] | We do not prove it is unique (after we do it is usually denoted [ilmath]-x[/ilmath], just that it exists The "forall [ilmath]x\in R[/ilmath] there exists" states that for a given [ilmath]x\in R[/ilmath] a y exists. Not a y exists for all [ilmath]x[/ilmath] |
Multiplication is associative | [math]\forall a,b,c\in R[(ab)c=a(bc)][/math] | |
Multiplication is distributive | [math]\forall a,b,c\in R[a(b+c)=ab+ac][/math] [math]\forall a,b,c\in R[(a+b)c = ac+bc][/math] |
Is a ring, which we write: [math](R,+:R\times R\rightarrow R,\times:R\times R\rightarrow R)[/math] but because Mathematicians are lazy we write simply:
- [math](R,+,\times)[/math]
Subring
If [ilmath](S,+,\times)[/ilmath] is a ring, and every element of [ilmath]S[/ilmath] is also in [ilmath]R[/ilmath] (for another ring [ilmath](R,+,\times)[/ilmath]) and the operations of addition and multiplication on [ilmath]S[/ilmath] are the same as those on [ilmath]R[/ilmath] (when restricted to [ilmath]S[/ilmath] of course) then we say "[ilmath]S[/ilmath] is a subring of [ilmath]R[/ilmath]"
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 [ilmath](G,+)[/ilmath] with another operation on [ilmath]G[/ilmath] called [ilmath]\times[/ilmath]
Properties
Name | Statement | Explanation |
---|---|---|
Commutative Ring | [math]\forall x,y\in R[xy=yx][/math] | 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 | [math]\exists e_\times\in R\forall x\in R[xe_\times=e_\times x=x][/math] | The existence of a multiplicative identity, once we have proved it is unique we often denote this "[ilmath]1[/ilmath]" |
Using properties
A commutative ring with unity is a ring with the additional properties of:
- [math]\forall x,y\in R[xy=yx][/math]
- [math]\exists e_\times\in R\forall x\in R[xe_\times=e_\times x=x][/math]
It is that simple.
Immediate theorems
Theorem: The additive identity of a ring [ilmath]R[/ilmath] is unique (and as such can be denoted [ilmath]0[/ilmath] unambiguously)
This is a classic "suppose there are two" proof, and we will do the same.
Suppose that [ilmath]0\in R[/ilmath] is such that [ilmath]\forall x\in R[0+x=x+0=x][/ilmath]
- Suppose that [ilmath]0'\in R[/ilmath] with [ilmath]0'\ne 0[/ilmath] and also such that: [ilmath]\forall x\in R[0'+x=x+0'=x][/ilmath]
We will show that [ilmath]0=0'[/ilmath], contradicting them being different! Thus showing there is no other "zero"
Proof:
- [math]0+0'=0[/math] by the property of [ilmath]0[/ilmath]
- [math]0+0'=0'+0[/math] by the commutivity of addition
- [math]0'+0=0'[/math] by the property of [ilmath]0'[/ilmath]
- Thus [math]0=0'[/math]
- This contradicts that [ilmath]0\ne 0'[/ilmath] so the claim they are distinct cannot be, we have only one "zero element", which herein we shall denote as "[ilmath]0[/ilmath]"
(Cancellation laws) Theorem: if [ilmath]a+c=b+c[/ilmath] then [ilmath]a=b[/ilmath] (and due to commutivity of addition [math]c+a=c+b\implies a=b[/math] too)
Suppose that [ilmath]a+c=b+c[/ilmath]
- By the additive inverse property, [math]\exists x\in R:c+x=0[/math]
- First notice that [math](a+c)+x=(b+c)+x[/math] (using [math]a+c=b+c[/math])
- Let us take [math](a+c)+x[/math]
- By associativity of addition, [math](a+c)+x=a+(c+x)=a+0=a[/math]
- Let us take [math](b+c)+x[/math]
- By associativity of addition, [math](b+c)+x=b+(c+x)=b+0=b[/math]
- Let us take [math](a+c)+x[/math]
- We see that [math]a=a+c+x=b+c+x=b[/math]
- First notice that [math](a+c)+x=(b+c)+x[/math] (using [math]a+c=b+c[/math])
- Which is indeed just [math]a=b[/math]
As claimed.
Note:
- Note that [math]c+a=b+c\implies a=b[/math], this can be proved identically to the above (but adding x to the left) or by:
- [math]c+a=a+c[/math] and </math>b+c=c+b</math> and then apply the above.
Theorem: The additive inverse of an element is unique (and herein, for a given [ilmath]x\in R[/ilmath] shall be denoted [ilmath]-x[/ilmath])
TODO:
Important theorems
These theorems are "two steps away" from the definitions if you will, they are not immediate things like "the identity is unique"
Theorem: [math]\forall x\in R[0x=x0=0][/math] - an interesting result, in line with what we expect from our number system
Let [ilmath]x\in R[/ilmath] be given.
- Proof of: [ilmath]x0=0[/ilmath]
- Note that [ilmath]x=x+0[/ilmath] then
- [ilmath]xx=x(x+0)=xx+x0[/ilmath] by distributivity
- Note that [ilmath]xx=xx+0[/ilmath] then
- [ilmath]xx+0=xx+x0[/ilmath]
- [ilmath]xx=x(x+0)=xx+x0[/ilmath] by distributivity
- By the cancellation laws: [ilmath]\implies 0=x0[/ilmath]
- So we have shown [ilmath]\forall x\in R[x0=0][/ilmath]
- Note that [ilmath]x=x+0[/ilmath] then
- Proof of: [ilmath]0x=0[/ilmath]
- Note that [ilmath]x=x+0[/ilmath] then
- [ilmath]xx=(x+0)x=xx+0x[/ilmath] by distributivity
- Note that [ilmath]xx=xx+0[/ilmath] then
- [ilmath]xx+0=xx+0x[/ilmath]
- [ilmath]xx=(x+0)x=xx+0x[/ilmath] by distributivity
- By the cancellation laws: [ilmath]\implies 0=0x[/ilmath]
- So we have shown [ilmath]\forall x\in R[0x=0][/ilmath]
- Note that [ilmath]x=x+0[/ilmath] then
- So [math]\forall x\in R[0x=0\wedge x0=0][/math] or simply [math]\forall x\in R[0x=x0=0][/math]
This completes the proof.
See next
See also
References
- ↑ Fundamentals of abstract algebra - an expanded version - Neal H. McCoy
- Refactoring
- Todo
- Definitions
- Abstract Algebra Definitions
- Abstract Algebra
- Ring Theory Definitions
- Ring Theory
- Pages requiring references
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- Theorems
- Theorems, lemmas and corollaries
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- Abstract Algebra Theorems, lemmas and corollaries
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- Ring Theory Theorems, lemmas and corollaries
- First-year friendly