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NCAlgebra :: NCRingMap

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

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.
--running: bergman -i /var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-12452-0/0.init -on-error exit --silent > /var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-12452-0/3.ter ... 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

     |  3   2            |
o6 = | a   b   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 :

For the programmer

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