Ring

Ring: Group + multiplication (see: http://en.wikipedia.org/wiki/Ring_%28mathematics%29) and the three elements it defines:

additive identity aka zero
addition
multiplication

Note, if you have distributive property, additive inverses, and multiplicative identity you can prove you have a commutative group under the ring:

(a + 1)*(b + 1) = a(b + 1) + (b + 1) 2. = ab + a + b + 1 3. or: 4. 5. = (a + 1)b + (a + 1) 6. = ab + b + a + 1 7. 8. So: ab + a + b + 1 == ab + b + a + 1 9. using the fact that -(ab) and -1 exist, we get: 10. a + b == b + a

Annotations: @implicitNotFound( ... )

Linear Supertypes

algebra.ring.Ring[T], Rng[T], AdditiveCommutativeGroup[T], Rig[T], MultiplicativeMonoid[T], Semiring[T], MultiplicativeSemigroup[T], AdditiveCommutativeMonoid[T], AdditiveCommutativeSemigroup[T], CommutativeGroup[T], CommutativeMonoid[T], CommutativeSemigroup[T], Group[T], AdditiveGroup[T], Monoid[T], AdditiveMonoid[T], Semigroup[T], AdditiveSemigroup[T], cats.kernel.Group[T], cats.kernel.Monoid[T], cats.kernel.Semigroup[T], Serializable, Serializable, AnyRef, Any

Known Subclasses

AbstractRing, AdjoinedUnitRing, ApplicativeRing, BigDecimalRing, BigIntRing, BooleanRing, ConstantRing, DoubleRing, EventuallyRing, FloatRing, FromAlgebraRing, IndexedSeqRing, IntRing, InvariantRing, JBoolRing, JDoubleRing, JFloatRing, JIntRing, JLongRing, JShortRing, LongRing, NumericRing, Product10Ring, Product11Ring, Product12Ring, Product13Ring, Product14Ring, Product15Ring, Product16Ring, Product17Ring, Product18Ring, Product19Ring, Product20Ring, Product21Ring, Product22Ring, Product2Ring, Product3Ring, Product4Ring, Product5Ring, Product6Ring, Product7Ring, Product8Ring, Product9Ring, ShortRing, StatisticsRing, Tuple10Ring, Tuple11Ring, Tuple12Ring, Tuple13Ring, Tuple14Ring, Tuple15Ring, Tuple16Ring, Tuple17Ring, Tuple18Ring, Tuple19Ring, Tuple20Ring, Tuple21Ring, Tuple22Ring, Tuple2Ring, Tuple3Ring, Tuple4Ring, Tuple5Ring, Tuple6Ring, Tuple7Ring, Tuple8Ring, Tuple9Ring, UnsafeFromAlgebraRig, UnsafeFromAlgebraRng

Abstract Value Members

abstract def negate(x: T): T

Definition Classes
AdditiveGroup
abstract def one: T

Definition Classes
Ring → MultiplicativeMonoid
abstract def plus(x: T, y: T): T

Definition Classes
AdditiveSemigroup
abstract def times(a: T, b: T): T

Definition Classes
Ring → MultiplicativeSemigroup
abstract def zero: T

Definition Classes
AdditiveMonoid

Concrete Value Members

final def !=(arg0: AnyRef): Boolean

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

Definition Classes
Any
final def ##(): Int

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

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

Definition Classes
Any
def additive: CommutativeGroup[T]

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

Definition Classes
Any
def assertNotZero(v: T): Unit

Definition Classes
Monoid
def clone(): AnyRef

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( ... )
def combine(l: T, r: T): T

Definition Classes
Semigroup → Semigroup
def combineAll(t: TraversableOnce[T]): T

Definition Classes
Monoid → Monoid
def combineAllOption(iter: TraversableOnce[T]): Option[T]

Definition Classes
Semigroup → Semigroup
def combineN(a: T, n: Int): T

Definition Classes
Group → Monoid → Semigroup
def empty: T

Definition Classes
Monoid → Monoid
final def eq(arg0: AnyRef): Boolean

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

Definition Classes
AnyRef → Any
def finalize(): Unit

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( classOf[java.lang.Throwable] )
def fromBigInt(n: BigInt): T

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

Definition Classes
Ring
final def getClass(): Class[_]

Definition Classes
AnyRef → Any
def hashCode(): Int

Definition Classes
AnyRef → Any
def inverse(v: T): T

Definition Classes
Group → Group
def isEmpty(a: T)(implicit ev: Eq[T]): Boolean

Definition Classes
Monoid
final def isInstanceOf[T0]: Boolean

Definition Classes
Any
def isNonZero(v: T): Boolean

Definition Classes
Monoid
def isOne(a: T)(implicit ev: Eq[T]): Boolean

Definition Classes
MultiplicativeMonoid
def isZero(a: T)(implicit ev: Eq[T]): Boolean

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

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

Definition Classes
MultiplicativeMonoid → MultiplicativeSemigroup
final def ne(arg0: AnyRef): Boolean

Definition Classes
AnyRef
def nonZeroOption(v: T): Option[T]

Definition Classes
Monoid
final def notify(): Unit

Definition Classes
AnyRef
final def notifyAll(): Unit

Definition Classes
AnyRef
def positivePow(a: T, n: Int): T

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

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

Definition Classes
MultiplicativeMonoid → MultiplicativeSemigroup
def product(iter: TraversableOnce[T]): T

Definition Classes
Ring → MultiplicativeMonoid
def remove(l: T, r: T): T

Definition Classes
Group → Group
def repeatedCombineN(a: T, n: Int): T

Attributes
protected[this]
Definition Classes
Semigroup
def sum(vs: TraversableOnce[T]): T

Definition Classes
Monoid → AdditiveMonoid
def sumN(a: T, n: Int): T

Definition Classes
AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
def sumOption(iter: TraversableOnce[T]): Option[T]

Returns an instance of T calculated by summing all instances in iter in one pass.
Returns an instance of T calculated by summing all instances in iter in one pass. Returns None if iter is empty, else Some[T].
iter
instances of T to be combined
returns
None if iter is empty, else an option value containing the summed T

Definition Classes
Semigroup
Note
Override if there is a faster way to compute this sum than iter.reduceLeftOption using plus.
final def synchronized[T0](arg0: ⇒ T0): T0

Definition Classes
AnyRef
def toString(): String

Definition Classes
AnyRef → Any
def tryProduct(as: TraversableOnce[T]): Option[T]

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

Definition Classes
AdditiveMonoid → AdditiveSemigroup
final def wait(): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )
final def wait(arg0: Long, arg1: Int): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )
final def wait(arg0: Long): Unit

Definition Classes
AnyRef
Annotations
@throws( ... )

trait Ring[T] extends Group[T] with CommutativeGroup[T] with algebra.ring.Ring[T]

Abstract Value Members

abstract def negate(x: T): T

abstract def one: T

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

abstract def times(a: T, b: T): T

abstract def zero: T

Concrete Value Members

final def !=(arg0: AnyRef): Boolean

final def !=(arg0: Any): Boolean

final def ##(): Int

final def ==(arg0: AnyRef): Boolean

final def ==(arg0: Any): Boolean

def additive: CommutativeGroup[T]

final def asInstanceOf[T0]: T0

def assertNotZero(v: T): Unit

def clone(): AnyRef

def combine(l: T, r: T): T

def combineAll(t: TraversableOnce[T]): T

def combineAllOption(iter: TraversableOnce[T]): Option[T]

def combineN(a: T, n: Int): T

def empty: T

final def eq(arg0: AnyRef): Boolean

def equals(arg0: Any): Boolean

def finalize(): Unit

def fromBigInt(n: BigInt): T

def fromInt(n: Int): T

final def getClass(): Class[_]

def hashCode(): Int

def inverse(v: T): T

def isEmpty(a: T)(implicit ev: Eq[T]): Boolean

final def isInstanceOf[T0]: Boolean

def isNonZero(v: T): Boolean

def isOne(a: T)(implicit ev: Eq[T]): Boolean

def isZero(a: T)(implicit ev: Eq[T]): Boolean

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

def multiplicative: algebra.Monoid[T]

final def ne(arg0: AnyRef): Boolean

def nonZeroOption(v: T): Option[T]

final def notify(): Unit

final def notifyAll(): Unit

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

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

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

def product(iter: TraversableOnce[T]): T

def remove(l: T, r: T): T

def repeatedCombineN(a: T, n: Int): T

def sum(vs: TraversableOnce[T]): T

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

def sumOption(iter: TraversableOnce[T]): Option[T]

final def synchronized[T0](arg0: ⇒ T0): T0

def toString(): String

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

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

final def wait(): Unit

final def wait(arg0: Long, arg1: Int): Unit

final def wait(arg0: Long): Unit

Inherited from algebra.ring.Ring[T]

Inherited from Rng[T]

Inherited from AdditiveCommutativeGroup[T]

Inherited from Rig[T]

Inherited from MultiplicativeMonoid[T]

Inherited from Semiring[T]

Inherited from MultiplicativeSemigroup[T]

Inherited from AdditiveCommutativeMonoid[T]

Inherited from AdditiveCommutativeSemigroup[T]

Inherited from CommutativeGroup[T]

Inherited from CommutativeMonoid[T]

Inherited from CommutativeSemigroup[T]

Inherited from Group[T]

Inherited from AdditiveGroup[T]

Inherited from Monoid[T]

Inherited from AdditiveMonoid[T]

Inherited from Semigroup[T]

Inherited from AdditiveSemigroup[T]

Inherited from cats.kernel.Group[T]

Inherited from cats.kernel.Monoid[T]

Inherited from cats.kernel.Semigroup[T]

Inherited from Serializable