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<head><title>RingElement / RingElement -- fraction</title>
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<div><h1>RingElement / RingElement -- fraction</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>f/g</tt></div>
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<li><span>Operator: <a href="__sl.html" title="a binary operator, usually used for division">/</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Ring__Element.html">ring element</a></span></span></li>
<li><span><tt>g</tt>, <span>a <a href="___Ring__Element.html">ring element</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Ring__Element.html">ring element</a></span>, the fraction f/g</span></li>
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<div class="single"><h2>Description</h2>
<div>If either f or g is in a base ring of the other, then that one is promoted so that both are elements in the same ring R.<p/>
The fraction will be an element of the fraction field, frac R, of R. If R is already a field, then this means that the fraction will be an element of R.<table class="examples"><tr><td><pre>i1 : 4/2
o1 = 2
o1 : QQ</pre>
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<table class="examples"><tr><td><pre>i2 : R = GF(9,Variable=>a);</pre>
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<tr><td><pre>i3 : (a/a^3) * a^2 == 1
o3 = true</pre>
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<table class="examples"><tr><td><pre>i4 : S = ZZ[a,b]
o4 = S
o4 : PolynomialRing</pre>
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<tr><td><pre>i5 : (a^6-b^6)/(a^9-b^9)
3 3
a + b
o5 = --------------
6 3 3 6
a + a b + b
o5 : frac(S)</pre>
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If the ring contains zero divisors, the fraction field is not defined. Macaulay2 will not inform you of this right away. However, if computation finds a zero-divisor, an error message is generated.<table class="examples"><tr><td><pre>i6 : A = ZZ/101[a,b]/(a*b)
o6 = A
o6 : QuotientRing</pre>
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<tr><td><pre>i7 : (a+b)/(a-b)
-b
o7 = --
b
o7 : frac(A)</pre>
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At this point, if one types <tt>a/b</tt>, then Macaulay2 would give an error saying that a zero divisor was found in the denominator.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></li>
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