Ring/New page

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Not to be confused with a ring of sets


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:
      1. [ilmath]\forall a,b\in R[a\oplus b=b\oplus a][/ilmath] - commutivity
  • [ilmath](R,\odot)[/ilmath] is a semigroup
    • Semigroup definition:
      1. [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:
      1. [ilmath]\forall a,b,c\in R[a\odot(b\oplus c)=(a\odot b)\oplus(a\odot c)][/ilmath] and
      2. [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:

  1. [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]
  2. [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


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


  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


  1. Fundamentals of Abstract Algebra - Neal H. McCoy