quotient rings

quotient rings

The usual notation is used to form quotient rings. For quotients of polynomial rings, a Groebner basis is computed and used to reduce ring elements to normal form after arithmetic operations.

i1 : R = ZZ/11



o1 = R



o1 : QuotientRing

i2 : 6_R + 7_R



o2 = 2



o2 : R

i3 : S = QQ[x,y,z]/(x^2-y, y^3-z)



o3 = S



o3 : QuotientRing

i4 : {1,x,x^2,x^3,x^4,x^5,x^6,x^7,x^8}



                     2     2

o4 = {1, x, y, x*y, y , x*y , z, x*z, y*z}



o4 : List

In the example above you might have wondered whether typing x would give an element of S or an element of QQ[x,y,z]. Our convention is that typing x gives an element of the last ring that has been assigned to a global variable. Here is another example.

i5 : T = ZZ/101[r,s,t]



o5 = T



o5 : PolynomialRing

i6 : T/(r^3+s^3+t^3)



           T

o6 = ------------

      3    3    3

     r  + s  + t



o6 : QuotientRing

i7 : r^3+s^3+t^3



      3    3    3

o7 = r  + s  + t



o7 : T

Notice that this time, the variables end up in the ring T, because we didn't assign the quotient ring to a global variable. The command use would install the variables for us, or we could assign the ring to a global variable.

i8 : U = ooo



o8 = U



o8 : QuotientRing

i9 : r^3+s^3+t^3



o9 = 0



o9 : U

The functions lift and promote can be used to transfer elements between the polynomial ring and its quotient ring.

i10 : lift(U_"r",T)



o10 = r



o10 : T

i11 : promote(T_"r",U)



o11 = r



o11 : U

A random element of degree n can be obtained with random.

i12 : random(2,S)



       2   5       8  2

o12 = y  + -*x*z - -*z

           2       9



o12 : S

In a program we can tell whether a ring is a quotient ring.

i13 : isQuotientRing ZZ



o13 = false

i14 : isQuotientRing S



o14 = true

We can recover the ring of which a given ring is a quotient.

i15 : ambient S



o15 = QQ [x, y, z]



o15 : PolynomialRing

We can also recover the coefficient ring, as we could for the original polynomial ring.

i16 : coefficientRing S



o16 = QQ



o16 : Ring



  --  the class of all rational numbers

Here's how we can tell whether the defining relations of a quotient ring were homogeneous.

i17 : isHomogeneous S



o17 = false

i18 : isHomogeneous U



o18 = true

We can obtain the characteristic of a ring with char.

i19 : char (ZZ/11)



o19 = 11

i20 : char S



o20 = 0

i21 : char U



o21 = 101

The presentation of the quotient ring can be obtained as a matrix with presentation.

i22 : presentation S



o22 = | x2-y y3-z |



                         1                  2

o22 : Matrix QQ [x, y, z]  <--- QQ [x, y, z]

If a quotient ring has redundant defining relations, a new ring can be made in which these are eliminated with trim.

i23 : R = ZZ/101[x,y,z]/(x-y,y-z,z-x)



o23 = R



o23 : QuotientRing

i24 : trim R



        ZZ

       --- [x, y, z]

       101

o24 = --------------

      (y - z, x - z)



o24 : QuotientRing

For more information see QuotientRing.