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<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="_ideals.html" title="">ideals</a> > <a href="_equality_spand_spcontainment.html" title="">equality and containment</a></div>
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<div><h1>equality and containment</h1>
<div>Equality and containment between two ideals in a polynomial ring (or quotient of a polynomial ring) is checked by comparing their respective Groebner bases.<h2>equal and not equal</h2>
Use <a href="__eq_eq.html" title="equality">Ideal == Ideal</a> to test if two ideals in the same ring are equal.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : I = ideal (a^2*b-c^2, a*b^2-d^3, c^5-d);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : J = ideal (a^2,b^2,c^2,d^2);
o3 : Ideal of R</pre>
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<tr><td><pre>i4 : I == J
o4 = false</pre>
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<tr><td><pre>i5 : I != J
o5 = true</pre>
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<h2>normal form with respect to a Groebner basis and membership</h2>
The function <a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">RingElement % Ideal</a> reduces an element with respect to a Groebner basis of the ideal.<table class="examples"><tr><td><pre>i6 : (1+a+a^3+a^4) % J
o6 = a + 1
o6 : R</pre>
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We can then test membership in the ideal by comparing the answer to 0 using <a href="__eq_eq.html" title="equality">==</a>.<table class="examples"><tr><td><pre>i7 : (1+a+a^3+a^4) % J == 0
o7 = false</pre>
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<tr><td><pre>i8 : a^4 % J == 0
o8 = true</pre>
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<h2>containment for two ideals</h2>
Containment for two ideals is tested using <a href="_is__Subset.html" title="whether one object is a subset of another">isSubset</a>.<table class="examples"><tr><td><pre>i9 : isSubset(I,J)
o9 = false</pre>
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<tr><td><pre>i10 : isSubset(I,I+J)
o10 = true</pre>
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<tr><td><pre>i11 : isSubset(I+J,I)
o11 = false</pre>
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<h2>ideal equal to 1 or 0</h2>
Use the expression <tt>I == 1</tt> to see if the ideal is equal to the ring. Use <tt>I == 0</tt> to see if the ideal is identically zero in the given ring.<table class="examples"><tr><td><pre>i12 : I = ideal (a^2-1,a^3+3);
o12 : Ideal of R</pre>
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<tr><td><pre>i13 : I == 1
o13 = true</pre>
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<tr><td><pre>i14 : S = R/I
o14 = S
o14 : QuotientRing</pre>
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<tr><td><pre>i15 : S == 0
o15 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="__eq_eq.html" title="equality">Ideal == Ideal</a> -- equality</span></li>
<li><span><a href="__eq_eq.html" title="equality">Ideal == ZZ</a> -- equality</span></li>
<li><span><a href="_!_eq.html" title="inequality">!=</a> -- inequality</span></li>
<li><span><a href="_methods_spfor_spnormal_spforms_spand_spremainder.html" title="calculate the normal form of ring elements and matrices">RingElement % Ideal</a> -- calculate the normal form of ring elements and matrices</span></li>
<li><span><a href="_is__Subset_lp__Ideal_cm__Ideal_rp.html" title="whether one object is a subset of another">isSubset(Ideal,Ideal)</a> -- whether one object is a subset of another</span></li>
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