trait Ring[T] extends Group[T] with CommutativeGroup[T] with algebra.ring.Ring[T]
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
Known Subclasses
Ordering
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Inherited
- Ring
- Ring
- Rng
- AdditiveCommutativeGroup
- Rig
- MultiplicativeMonoid
- Semiring
- MultiplicativeSemigroup
- AdditiveCommutativeMonoid
- AdditiveCommutativeSemigroup
- CommutativeGroup
- CommutativeMonoid
- CommutativeSemigroup
- Group
- AdditiveGroup
- Monoid
- AdditiveMonoid
- Semigroup
- AdditiveSemigroup
- Group
- Monoid
- Semigroup
- Serializable
- AnyRef
- Any
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Visibility
- Public
- Protected
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: Any): Boolean
- Definition Classes
- AnyRef → Any
- final def ##: Int
- Definition Classes
- AnyRef → Any
- final def ==(arg0: Any): Boolean
- Definition Classes
- AnyRef → 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[lang]
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.CloneNotSupportedException]) @native()
- 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: AnyRef): Boolean
- Definition Classes
- AnyRef → Any
- def finalize(): Unit
- Attributes
- protected[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[_ <: AnyRef]
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
- def hashCode(): Int
- Definition Classes
- AnyRef → Any
- Annotations
- @native()
- 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
- Annotations
- @native()
- final def notifyAll(): Unit
- Definition Classes
- AnyRef
- Annotations
- @native()
- 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
Tcalculated by summing all instances initerin one pass.Returns an instance of
Tcalculated by summing all instances initerin one pass. ReturnsNoneifiteris empty, elseSome[T].- iter
instances of
Tto be combined- returns
Noneifiteris empty, else an option value containing the summedT
- 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(classOf[java.lang.InterruptedException])
- final def wait(arg0: Long, arg1: Int): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.InterruptedException])
- final def wait(arg0: Long): Unit
- Definition Classes
- AnyRef
- Annotations
- @throws(classOf[java.lang.InterruptedException]) @native()