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<head><title>Singular Book 1.8.13 -- quotient of ideals</title>
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<div><h1>Singular Book 1.8.13 -- quotient of ideals</h1>
<div><table class="examples"><tr><td><pre>i1 : A = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : I1 = ideal(x,y);
o2 : Ideal of A</pre>
</td></tr>
<tr><td><pre>i3 : I2 = ideal(y^2,z);
o3 : Ideal of A</pre>
</td></tr>
<tr><td><pre>i4 : I1 : I2
o4 = ideal (y, x)
o4 : Ideal of A</pre>
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The quotient function is the same as the colon operator, except that optional arguments may be given.<table class="examples"><tr><td><pre>i5 : quotient(I1,I2)
o5 = ideal (y, x)
o5 : Ideal of A</pre>
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Now we use the method described in Lemma 1.8.12 in the Singular book.<table class="examples"><tr><td><pre>i6 : J1 = intersect(I1,ideal(I2_0))
2
o6 = ideal(y )
o6 : Ideal of A</pre>
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<tr><td><pre>i7 : J2 = intersect(I1,ideal(I2_1))
o7 = ideal (y*z, x*z)
o7 : Ideal of A</pre>
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Now divide each generator of J1 by x, and each generator of J2 by y. Notice that division uses two slashes. Using only one slash gives the quotient in the fraction ring.<table class="examples"><tr><td><pre>i8 : K1 = ideal(J1_0//I2_0)
o8 = ideal 1
o8 : Ideal of A</pre>
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<tr><td><pre>i9 : K2 = ideal(J2_0//I2_1, J2_1//I2_1)
o9 = ideal (y, x)
o9 : Ideal of A</pre>
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<tr><td><pre>i10 : intersect(K1,K2)
o10 = ideal (y, x)
o10 : Ideal of A</pre>
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