Latest revision as of 05:02, 16 October 2016

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]):

1. [ilmath]\oplus:R\times R\rightarrow R[/ilmath] - called "addition", [ilmath]\oplus:(a,b)\mapsto a\oplus b[/ilmath]
2. [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:
    1. [ilmath]\forall a,b,c\in R[(a\oplus b)\oplus c=a\oplus(b\oplus c)][/ilmath] - associativity
    2. [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]
    3. [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:
    4. [ilmath]\forall a,b\in R[a\oplus b=b\oplus a][/ilmath] - commutivity
• [ilmath](R,\odot)[/ilmath] is a semigroup
  • Semigroup definition:
    5. [ilmath]\forall a,b,c\in R[(a\odot b)\odot c=a\odot(b\odot c)][/ilmath]
• 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:
    6. [ilmath]\forall a,b,c\in R[a\odot(b\oplus c)=(a\odot b)\oplus(a\odot c)][/ilmath] and
    7. [ilmath]\forall a,b,c\in R[(a+b)c=ac+bc][/ilmath]

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:

8. [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]
9. [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]}:

1. Ring - properties 1-7
2. Ring with unity (AKA: u-ring, ring with identity) - properties 1-8
3. Commutative ring (AKA: c-ring) - properties 1-7 and 9
4. 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
• Unit of a ring
• Division ring
• Ring ideal
• Quotient ring
• Fundamental ring homomorphism theorem
• Ring isomorphism theorems
• Module

Notes

1. ↑ So we could have [ilmath]0_R=1_R[/ilmath] or we could have [ilmath]0_R\ne 1_R[/ilmath]
2. ↑ there is only one inverse
3. ↑ there is only one inverse for an element
4. ↑ For multiplicative notation we'd use [ilmath]a^{-1} [/ilmath]
5. ↑ field, integral domain are also all rings, there's like 6 kinds. We call "Elementary ring" just the ones listed

References

1. ↑ Fundamentals of Abstract Algebra - Neal H. McCoy

OLD PAGE

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]:

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

Using properties

A commutative ring with unity is a ring with the additional properties of:

1. [math]\forall x,y\in R[xy=yx][/math]
2. [math]\exists e_\times\in R\forall x\in R[xe_\times=e_\times x=x][/math]

It is that simple.

Immediate theorems

Important theorems

These theorems are "two steps away" from the definitions if you will, they are not immediate things like "the identity is unique"

See next

• Examples of rings
• Direct sum of rings
• Integral domain

See also

• Group
• Category for various types of rings

References

1. ↑ Fundamentals of abstract algebra - an expanded version - Neal H. McCoy