NCRingMap -- Type of a map to or from a noncommutative ring.

Description

As in the commutative case, a map F:R->S where R or S is an NCRing is specified by giving the images in S of the variables of R. The target map is given first.

Common ways to make (and use) an NCRingMap include

ncMap(Ring,NCRing,List) -- Make a map to or from an NCRing
ncMap(NCRing,Ring,List) -- Make a map to or from an NCRing
ncMap(NCRing,NCRing,List) -- Make a map to or from an NCRing
normalElements(NCRingMap,ZZ) -- Finds elements normalized by a ring map
oreExtension(NCRing,NCRingMap,NCRingMap,NCRingElement) -- Creates an Ore extension of a noncommutative ring
oreExtension(NCRing,NCRingMap,NCRingMap,Symbol) -- Creates an Ore extension of a noncommutative ring
oreExtension(NCRing,NCRingMap,NCRingElement) -- Creates an Ore extension of a noncommutative ring
oreExtension(NCRing,NCRingMap,Symbol) -- Creates an Ore extension of a noncommutative ring
oreIdeal(NCRing,NCRingMap,NCRingMap,NCRingElement) -- Creates the defining ideal of an Ore extension of a noncommutative ring
oreIdeal(NCRing,NCRingMap,NCRingMap,Symbol) -- Creates the defining ideal of an Ore extension of a noncommutative ring
oreIdeal(NCRing,NCRingMap,NCRingElement) -- Creates the defining ideal of an Ore extension of a noncommutative ring
oreIdeal(NCRing,NCRingMap,Symbol) -- Creates the defining ideal of an Ore extension of a noncommutative ring

Common ways to get information about NCRingMaps Code UL {TO (source,NCRingMap), TO (target,NCRingMap), TO (matrix,NCRingMap), TO (isWellDefined,NCRingMap), TO (isHomogeneous,NCRingMap), TO (symbol _, NCRingMap, ZZ)} Text Common operations involving NCRingMaps Code UL {TO (ambient,NCRingMap), TO (symbol /,List,NCRingMap), TO (symbol SPACE, NCRingMap, NCRingElement), TO (symbol SPACE, NCRingMap, RingElement), TO (symbol SPACE, NCRingMap, NCMatrix), }

i1 : A = skewPolynomialRing(QQ,(-1)_QQ,{w,x,y,z})
--Calling Bergman for NCGB calculation.
Complete!

o1 = A

o1 : NCQuotientRing

i2 : B = QQ{a,b,c}

o2 = B

o2 : NCPolynomialRing

i3 : f = ncMap(B,A,{a^3,b^2,a+b,a-b})

o3 = NCRingMap B <--- A

o3 : NCRingMap

i4 : target f

o4 = B

o4 : NCPolynomialRing

i5 : source f

o5 = A

o5 : NCQuotientRing

i6 : matrix f

o6 = | a^3 b^2 b+a -b+a |

o6 : NCMatrix

Note that NCRingMaps need not be well-defined or homogeneous. Apply a function to an element or a matrix using the usual function notation. NCRingMaps are linear and multiplicative by definition.

i7 : f(w*x+2*y)

         3 2
o7 = 2b+a b +2a

o7 : B

i8 : isWellDefined f

o8 = false

i9 : isHomogeneous f

o9 = false

The user has the option to define an NCRingMap to be a derivation. Of course, such a map must have the same source and target.

i10 : g = ncMap(B,B,{a*b,b^2,c*a*c},Derivation=>true)

o10 = NCRingMap B <--- B

o10 : NCRingMap

i11 : g(a*b)==g(a)*b+a*g(b)

o11 = true

i12 : g(promote(1,B))

o12 = 0

o12 : B

i13 : g(c*a+2*b)

                 2
o13 = caca+cab+2b

o13 : B

Methods that use an object of class NCRingMap :

ambient(NCRingMap) -- Extends an NCRingMap to the ambient ring of the source.
gddKernel(ZZ,NCRingMap) -- see gddKernel -- Computes a homogeneous generating set of the kernel of a ring map.
isHomogeneous(NCRingMap) -- Determines if an NCRingMap preserves the natural grading
isWellDefined(NCRingMap) -- Determines if an NCRingMap is well-defined.
kernelComponent(ZZ,NCRingMap) -- see kernelComponent -- Computes a basis of the kernel of a ring map in a specified degree.
List / NCRingMap -- Applies an NCRingMap to each element of a list
matrix(NCRingMap) -- An NCMatrix associated to an NCRingMap.
NCRingMap + NCRingMap -- Basic operations with NCRingMaps
NCRingMap ^ ZZ -- see NCRingMap + NCRingMap -- Basic operations with NCRingMaps
QQ * NCRingMap -- see NCRingMap + NCRingMap -- Basic operations with NCRingMaps
RingElement * NCRingMap -- see NCRingMap + NCRingMap -- Basic operations with NCRingMaps
ZZ * NCRingMap -- see NCRingMap + NCRingMap -- Basic operations with NCRingMaps
NCRingMap @@ NCRingMap -- Compose two NCRingMaps
NCRingMap _ ZZ -- Matrix of one homogeneous component of an NCRingMap
NCRingMap NCGroebnerBasis -- see NCRingMap NCIdeal -- Apply a ring map to the generators of an ideal
NCRingMap NCIdeal -- Apply a ring map to the generators of an ideal
NCRingMap NCMatrix -- see NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
NCRingMap RingElement -- see NCRingMap NCRingElement -- Apply an NCRingMap to an element or matrix
normalElements(NCRingMap,ZZ) -- Finds elements normalized by a ring map
oreExtension(NCRing,NCRingMap,NCRingElement) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
oreExtension(NCRing,NCRingMap,NCRingMap,NCRingElement) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
oreExtension(NCRing,NCRingMap,NCRingMap,Symbol) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
oreExtension(NCRing,NCRingMap,Symbol) -- see oreExtension -- Creates an Ore extension of a noncommutative ring
oreIdeal(NCRing,NCRingMap,NCRingElement) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
oreIdeal(NCRing,NCRingMap,NCRingMap,NCRingElement) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
oreIdeal(NCRing,NCRingMap,NCRingMap,Symbol) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
oreIdeal(NCRing,NCRingMap,Symbol) -- see oreIdeal -- Creates the defining ideal of an Ore extension of a noncommutative ring
source(NCRingMap) -- Source of a map
target(NCRingMap) -- Target of a map

For the programmer

The object NCRingMap is a type, with ancestor classes HashTable < Thing.