Non-associative ring

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Definition

A non-associative ring, more properly called a possibly non-associative ring or a not necessarily associative ring, is defined as 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