# 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