quotient rings

The usual notation is used to form quotient rings. For quotients of polynomial rings, a Gröbner 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 substitute can be used to transfer elements between the polynomial ring and its quotient ring.

i10 : lift("r"_U,T)

o10 = r

o10 : T

i11 : substitute("r"_T,U)

o11 = r

o11 : U

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

i12 : random(2,S)

      9      9 2   1      1       2
o12 = -x*y + -y  + -x*z + -y*z + z
      2      4     2      2

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..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

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..z])  <-- (QQ[x..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..z]
         101
o24 = --------------
      (y - z, x - z)

o24 : QuotientRing

For more information see QuotientRing.