# EuclidianRing

### Related Doc: package algebra

#### trait EuclidianRing extends CommutativeRing with RingWithDivision

Euclidian rings extend rings with operations for division and remainder, with the Euclidian definition: `plus(mul(div(s, t), t), mod(s, t)) === s`, with `f(mod(s, t)) in [0, abs(t))` for some appropriate embedding into real numbers.

Linear Supertypes
Known Subclasses
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Inherited
1. EuclidianRing
2. RingWithDivision
3. CommutativeRing
4. CommutativePseudoRing
5. Ring
6. PseudoRing
7. AnyRef
8. Any
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### Abstract Value Members

1. #### abstract def div(s: ITerm, t: ITerm): ITerm

Euclidian division

Euclidian division

Definition Classes
EuclidianRingRingWithDivision
2. #### abstract val dom: Sort

Domain of the ring

Domain of the ring

Definition Classes
PseudoRing
3. #### abstract def int2ring(s: ITerm): ITerm

Conversion of an integer term to a ring term

Conversion of an integer term to a ring term

Definition Classes
PseudoRing
4. #### abstract def minus(s: ITerm): ITerm

Definition Classes
PseudoRing
5. #### abstract def mod(s: ITerm, t: ITerm): ITerm

Euclidian remainder

6. #### abstract def mul(s: ITerm, t: ITerm): ITerm

Ring multiplication

Ring multiplication

Definition Classes
PseudoRing
7. #### abstract def one: ITerm

The one element of this ring

The one element of this ring

Definition Classes
PseudoRing
8. #### abstract def plus(s: ITerm, t: ITerm): ITerm

Definition Classes
PseudoRing
9. #### abstract def zero: ITerm

The zero element of this ring

The zero element of this ring

Definition Classes
PseudoRing

### Concrete Value Members

1. #### final def !=(arg0: Any): Boolean

Definition Classes
AnyRef → Any
2. #### final def ##(): Int

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

Definition Classes
AnyRef → Any
4. #### def additiveGroup: Group with Abelian with SymbolicTimes

Addition gives rise to an Abelian group

Addition gives rise to an Abelian group

Definition Classes
PseudoRing
5. #### final def asInstanceOf[T0]: T0

Definition Classes
Any
6. #### def clone(): AnyRef

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( ... )
7. #### final def eq(arg0: AnyRef): Boolean

Definition Classes
AnyRef
8. #### def equals(arg0: Any): Boolean

Definition Classes
AnyRef → Any
9. #### def finalize(): Unit

Attributes
protected[java.lang]
Definition Classes
AnyRef
Annotations
@throws( classOf[java.lang.Throwable] )
10. #### final def getClass(): Class[_]

Definition Classes
AnyRef → Any
11. #### def hashCode(): Int

Definition Classes
AnyRef → Any
12. #### final def isInstanceOf[T0]: Boolean

Definition Classes
Any
13. #### def minus(s: ITerm, t: ITerm): ITerm

Difference between two terms

Difference between two terms

Definition Classes
PseudoRing
14. #### def multiplicativeMonoid: Monoid with Abelian

Multiplication gives rise to an Abelian monoid

Multiplication gives rise to an Abelian monoid

Definition Classes
CommutativeRingRing
15. #### final def ne(arg0: AnyRef): Boolean

Definition Classes
AnyRef
16. #### final def notify(): Unit

Definition Classes
AnyRef
17. #### final def notifyAll(): Unit

Definition Classes
AnyRef
18. #### def product(terms: ITerm*): ITerm

N-ary sums

N-ary sums

Definition Classes
PseudoRing
19. #### def summation(terms: ITerm*): ITerm

N-ary sums

N-ary sums

Definition Classes
PseudoRing
20. #### final def synchronized[T0](arg0: ⇒ T0): T0

Definition Classes
AnyRef
21. #### def times(num: IdealInt, s: ITerm): ITerm

`num * s`

`num * s`

Definition Classes
PseudoRing
22. #### def toString(): String

Definition Classes
PseudoRing → AnyRef → Any
23. #### final def wait(): Unit

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

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

Definition Classes
AnyRef
Annotations
@throws( ... )