Associative ring

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A ring is a set R equipped with the following operations:

  • An infix binary operation +, called addition.
  • A prefix unary operation -, called the negative.
  • A constant element 0, called zero.
  • A binary operation *, called the multiplication.

satisfying the following compatibility conditions:

  • R forms an abelian group with group operation +, inverse operation -, and identity element 0.
  • R satisfies the two distributivity laws:
    • a * (b + c) = (a * b) + (a * c) \ \forall \ a,b,c \in R
    • (a + b) * c = (a * c) + (b * c) \ \forall \ a,b,c \in R
  • Associativity: a * (b * c) = (a * b) * c \ \forall \ a,b,c \in R

However, in many contexts, it is useful to study the situation where * is possibly non-associative, i.e., we want to remove the last condition from the definition. We use the term non-associative ring for a ring that is not necessarily associative. Note that associative rings are non-associative rings by this definition.

A unital ring or unitary ring is a ring with an identity for multiplication, denoted 1. A commutative ring is a ring where the multiplication is commutative. A commutative unital ring is a ring where the multiplication is both commutative and has a unit.