DivisionRing overview

See {@link DivisionRing} for details

Added in v1.0.0

Table of contents

• utils
  • DivisionRing (interface)
  • divL
  • divR

utils

DivisionRing (interface)

The DivisionRing class is for non-zero rings in which every non-zero element has a multiplicative inverse. Division rings are sometimes also called skew fields.

Instances must satisfy the following laws in addition to the {@link Ring} laws:

• Non-zero ring: one /= zero
• Non-zero multiplicative inverse: recip a * a = a * recip a = one for all non-zero a

The result of recip zero is left undefined; individual instances may choose how to handle this case.

If a type has both DivisionRing and {@link CommutativeRing} instances, then it is a field and should have a Field instance.

Signature

export interface DivisionRing<A> extends Ring<A> {
  readonly recip: (a: NonZero<A>) => A
}

Added in v1.0.0

divL

Left division, defined as divL(a, b) = recip(b * a). Left and right division are distinct in this module because a DivisionRing is not necessarily commutative.

If the type a is also a EuclideanRing, then this function is equivalent to div from the EuclideanRing class. When working abstractly, div should generally be preferred, unless you know that you need your code to work with noncommutative rings.

Signature

export declare function divL<A>(DR: DivisionRing<A>): (a1: A, a2: NonZero<A>) => A

Added in v1.0.0

divR

Right division, defined as rightDiv a b = a * recip b. Left and right division are distinct in this module because a DivisionRing is not necessarily commutative.

If the type a is also a EuclideanRing, then this function is equivalent to div from the EuclideanRing class. When working abstractly, div should generally be preferred, unless you know that you need your code to work with noncommutative rings.

Signature

export declare function divR<A>(DR: DivisionRing<A>): (a1: A, a2: NonZero<A>) => A

Added in v1.0.0