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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_rings.html" title="">rings</a> > <a href="_quotient_springs.html" title="">quotient rings</a></div>
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<div><h1>quotient rings</h1>
<div>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.<table class="examples"><tr><td><pre>i1 : R = ZZ/11
o1 = R
o1 : QuotientRing</pre>
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<tr><td><pre>i2 : 6_R + 7_R
o2 = 2
o2 : R</pre>
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<table class="examples"><tr><td><pre>i3 : S = QQ[x,y,z]/(x^2-y, y^3-z)
o3 = S
o3 : QuotientRing</pre>
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<tr><td><pre>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</pre>
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In the example above you might have wondered whether typing <tt>x</tt> would give an element of <tt>S</tt> or an element of <tt>QQ[x,y,z]</tt>. Our convention is that typing <tt>x</tt> gives an element of the last ring that has been assigned to a global variable. Here is another example.<table class="examples"><tr><td><pre>i5 : T = ZZ/101[r,s,t]
o5 = T
o5 : PolynomialRing</pre>
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<tr><td><pre>i6 : T/(r^3+s^3+t^3)
T
o6 = ------------
3 3 3
r + s + t
o6 : QuotientRing</pre>
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<tr><td><pre>i7 : r^3+s^3+t^3
3 3 3
o7 = r + s + t
o7 : T</pre>
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Notice that this time, the variables end up in the ring <tt>T</tt>, because we didn't assign the quotient ring to a global variable. The command <a href="_use.html" title="install or activate object">use</a> would install the variables for us, or we could assign the ring to a global variable.<table class="examples"><tr><td><pre>i8 : U = ooo
o8 = U
o8 : QuotientRing</pre>
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<tr><td><pre>i9 : r^3+s^3+t^3
o9 = 0
o9 : U</pre>
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The functions <a href="_lift.html" title="lift to another ring">lift</a> and <a href="_substitute.html" title="substituting values for variables">substitute</a> can be used to transfer elements between the polynomial ring and its quotient ring.<table class="examples"><tr><td><pre>i10 : lift(U_"r",T)
o10 = r
o10 : T</pre>
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<tr><td><pre>i11 : substitute(T_"r",U)
o11 = r
o11 : U</pre>
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A random element of degree <tt>n</tt> can be obtained with <a href="_random.html" title="get a random element">random</a>.<table class="examples"><tr><td><pre>i12 : random(2,S)
1 1 2 4 7 2
o12 = -x*y + -y + -x*z + y*z + -z
4 2 5 6
o12 : S</pre>
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In a program we can tell whether a ring is a quotient ring.<table class="examples"><tr><td><pre>i13 : isQuotientRing ZZ
o13 = false</pre>
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<tr><td><pre>i14 : isQuotientRing S
o14 = true</pre>
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We can recover the ring of which a given ring is a quotient.<table class="examples"><tr><td><pre>i15 : ambient S
o15 = QQ[x, y, z]
o15 : PolynomialRing</pre>
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We can also recover the coefficient ring, as we could for the original polynomial ring.<table class="examples"><tr><td><pre>i16 : coefficientRing S
o16 = QQ
o16 : Ring</pre>
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Here's how we can tell whether the defining relations of a quotient ring were homogeneous.<table class="examples"><tr><td><pre>i17 : isHomogeneous S
o17 = false</pre>
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<tr><td><pre>i18 : isHomogeneous U
o18 = true</pre>
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We can obtain the characteristic of a ring with <a href="_char.html" title="computes the characteristic of the ring or field">char</a>.<table class="examples"><tr><td><pre>i19 : char (ZZ/11)
o19 = 11</pre>
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<tr><td><pre>i20 : char S
o20 = 0</pre>
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<tr><td><pre>i21 : char U
o21 = 101</pre>
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The presentation of the quotient ring can be obtained as a matrix with <a href="_presentation.html" title="presentation of a module or ring">presentation</a>.<table class="examples"><tr><td><pre>i22 : presentation S
o22 = | x2-y y3-z |
1 2
o22 : Matrix (QQ[x, y, z]) <--- (QQ[x, y, z])</pre>
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If a quotient ring has redundant defining relations, a new ring can be made in which these are eliminated with <a href="_trim.html" title="minimize generators and relations">trim</a>.<table class="examples"><tr><td><pre>i23 : R = ZZ/101[x,y,z]/(x-y,y-z,z-x)
o23 = R
o23 : QuotientRing</pre>
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<tr><td><pre>i24 : trim R
ZZ
---[x, y, z]
101
o24 = --------------
(y - z, x - z)
o24 : QuotientRing</pre>
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For more information see <a href="___Quotient__Ring.html" title="the class of all quotient rings">QuotientRing</a>.</div>
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