RingAssociativeAlgebra

A RingAssociativeAlgebra is a R-module that is also a Ring. An example is the Gaussian numbers, the quaternions, etc...

The scalar multiplication satisfies, for r in R, and x, y in V:

1. r *: (x * y) = (r *: x) * y = x * (r *: y)

TODO: verify the definition, in particular the requirements for Ring[V] (and not Rng[V])

Linear Supertypes

algebra.ring.Ring[V], algebra.ring.Rng[V], algebra.ring.Rig[V], algebra.ring.MultiplicativeMonoid[V], algebra.ring.Semiring[V], algebra.ring.MultiplicativeSemigroup[V], CModule[V, R], RightModule[V, R], LeftModule[V, R], AdditiveCommutativeGroup[V], AdditiveCommutativeMonoid[V], AdditiveCommutativeSemigroup[V], algebra.ring.AdditiveGroup[V], algebra.ring.AdditiveMonoid[V], algebra.ring.AdditiveSemigroup[V], Serializable, Serializable, Any

Known Subclasses

FieldAssociativeAlgebra, PolynomialOverCRing, PolynomialOverField, ZAlgebra

Abstract Value Members

abstract def getClass(): Class[_]

Definition Classes
Any
abstract def negate(x: V): V

Definition Classes
AdditiveGroup
abstract def one: V

Definition Classes
MultiplicativeMonoid
abstract def plus(x: V, y: V): V

Definition Classes
AdditiveSemigroup
implicit abstract def scalar: CRing[R]

Definition Classes
CModule → RightModule → LeftModule
abstract def times(x: V, y: V): V

Definition Classes
MultiplicativeSemigroup
abstract def timesl(r: R, v: V): V

Definition Classes
LeftModule
abstract def zero: V

Definition Classes
AdditiveMonoid

Concrete Value Members

final def !=(arg0: Any): Boolean

Definition Classes
Any
final def ##(): Int

Definition Classes
Any
final def ==(arg0: Any): Boolean

Definition Classes
Any
def additive: CommutativeGroup[V]

Definition Classes
AdditiveCommutativeGroup → AdditiveCommutativeMonoid → AdditiveCommutativeSemigroup → AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
final def asInstanceOf[T0]: T0

Definition Classes
Any
def equals(arg0: Any): Boolean

Definition Classes
Any
def fromBigInt(n: BigInt): V

Definition Classes
Ring
def fromInt(n: Int): V

Definition Classes
Ring
def hashCode(): Int

Definition Classes
Any
final def isInstanceOf[T0]: Boolean

Definition Classes
Any
def isOne(a: V)(implicit ev: algebra.Eq[V]): Boolean

Definition Classes
MultiplicativeMonoid
def isZero(a: V)(implicit ev: algebra.Eq[V]): Boolean

Definition Classes
AdditiveMonoid
def minus(x: V, y: V): V

Definition Classes
AdditiveGroup
def multiplicative: algebra.Monoid[V]

Definition Classes
MultiplicativeMonoid → MultiplicativeSemigroup
def positivePow(a: V, n: Int): V

Attributes
protected[this]
Definition Classes
MultiplicativeSemigroup
def positiveSumN(a: V, n: Int): V

Attributes
protected[this]
Definition Classes
AdditiveSemigroup
def pow(a: V, n: Int): V

Definition Classes
MultiplicativeMonoid → MultiplicativeSemigroup
def product(as: TraversableOnce[V]): V

Definition Classes
MultiplicativeMonoid
def sum(as: TraversableOnce[V]): V

Definition Classes
AdditiveMonoid
def sumN(a: V, n: Int): V

Definition Classes
AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
def timesr(v: V, r: R): V

Definition Classes
CModule → RightModule
def toString(): String

Definition Classes
Any
def tryProduct(as: TraversableOnce[V]): Option[V]

Definition Classes
MultiplicativeMonoid → MultiplicativeSemigroup
def trySum(as: TraversableOnce[V]): Option[V]

Definition Classes
AdditiveMonoid → AdditiveSemigroup

Inherited from algebra.ring.Ring[V]

Inherited from algebra.ring.Rng[V]

Inherited from algebra.ring.Rig[V]

Related Docs: object RingAssociativeAlgebra | package algebra

trait RingAssociativeAlgebra[V, R] extends CModule[V, R] with Ring[V]

Abstract Value Members

abstract def getClass(): Class[_]

abstract def negate(x: V): V

abstract def one: V

abstract def plus(x: V, y: V): V

implicit abstract def scalar: CRing[R]

abstract def times(x: V, y: V): V

abstract def timesl(r: R, v: V): V

abstract def zero: V

Concrete Value Members

final def !=(arg0: Any): Boolean

final def ##(): Int

final def ==(arg0: Any): Boolean

def additive: CommutativeGroup[V]

final def asInstanceOf[T0]: T0

def equals(arg0: Any): Boolean

def fromBigInt(n: BigInt): V

def fromInt(n: Int): V

def hashCode(): Int

final def isInstanceOf[T0]: Boolean

def isOne(a: V)(implicit ev: algebra.Eq[V]): Boolean

def isZero(a: V)(implicit ev: algebra.Eq[V]): Boolean

def minus(x: V, y: V): V

def multiplicative: algebra.Monoid[V]

def positivePow(a: V, n: Int): V

def positiveSumN(a: V, n: Int): V

def pow(a: V, n: Int): V

def product(as: TraversableOnce[V]): V

def sum(as: TraversableOnce[V]): V

def sumN(a: V, n: Int): V

def timesr(v: V, r: R): V

def toString(): String

def tryProduct(as: TraversableOnce[V]): Option[V]

def trySum(as: TraversableOnce[V]): Option[V]

Inherited from algebra.ring.Ring[V]

Inherited from algebra.ring.Rng[V]

Inherited from algebra.ring.Rig[V]

Inherited from algebra.ring.MultiplicativeMonoid[V]

Inherited from algebra.ring.Semiring[V]

Inherited from algebra.ring.MultiplicativeSemigroup[V]

Inherited from CModule[V, R]

Inherited from RightModule[V, R]

Inherited from LeftModule[V, R]

Inherited from AdditiveCommutativeGroup[V]

Inherited from AdditiveCommutativeMonoid[V]

Inherited from AdditiveCommutativeSemigroup[V]

Inherited from algebra.ring.AdditiveGroup[V]

Inherited from algebra.ring.AdditiveMonoid[V]

Inherited from algebra.ring.AdditiveSemigroup[V]

Inherited from Serializable

Inherited from Serializable

Inherited from Any

Ungrouped