# Difference between revisions of "ApCoCoA-1:NC.Intersection"

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− | <em> | + | <em>Proposition (Intersection of Two Ideals):</em> Let <tt>G_I</tt> and <tt>G_J</tt> be two sets of non-zero polynomials in the non-commutative polynomial ring <tt>K<x[1],...,x[n]></tt>, and let <tt>I</tt> and <tt>J</tt> be two ideals generated by <tt>G_I</tt> and <tt>G_J</tt>, respectively. We choose a new indeterminate <tt>y</tt>, and form the free monoid ring <tt>K<y,x[1],...,x[n]></tt>. Furthermore, let <tt>N</tt> be the ideal generated by the union of <tt>{yf: f in G_I}</tt> and <tt>{(1-y)g: g in G_J}</tt>, and let <tt>C</tt> be the ideal generated by the set <tt>{yx[1]-x[1]y,...,yx[n]-x[n]y}</tt> of commutators. Then we have the intersection of <tt>I</tt> and <tt>J</tt> is equal to the intersection of <tt>N+C</tt> and <tt>K<x[1],...,x[n]></tt>. |

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-- Let I be the ideal generated by G_I={xy+z,yz+x}, and J be the ideal generated by G_J={yz+x, zx+y}. | -- Let I be the ideal generated by G_I={xy+z,yz+x}, and J be the ideal generated by G_J={yz+x, zx+y}. |

## Revision as of 14:57, 9 May 2013

## NC.Intersection

Intersection of two finitely generated two-sided ideals in a non-commutative polynomial ring.

### Description

*Proposition (Intersection of Two Ideals):* Let `G_I` and `G_J` be two sets of non-zero polynomials in the non-commutative polynomial ring `K<x[1],...,x[n]>`, and let `I` and `J` be two ideals generated by `G_I` and `G_J`, respectively. We choose a new indeterminate `y`, and form the free monoid ring `K<y,x[1],...,x[n]>`. Furthermore, let `N` be the ideal generated by the union of `{yf: f in G_I}` and `{(1-y)g: g in G_J}`, and let `C` be the ideal generated by the set `{yx[1]-x[1]y,...,yx[n]-x[n]y}` of commutators. Then we have the intersection of `I` and `J` is equal to the intersection of `N+C` and `K<x[1],...,x[n]>`.

#### Example

-- Let I be the ideal generated by G_I={xy+z,yz+x}, and J be the ideal generated by G_J={yz+x, zx+y}. -- We compute the intersection of I and J as follows. Use QQ[t,x,y,z]; NC.SetOrdering("ELIM"); -- Choose an elimination word ordering for t F1 := [[x,y], [z]]; -- xy+z F2 := [[y,z], [x]]; -- yz+x G1 := [[y,z], [x]]; -- yz+x G2 := [[z,x], [y]]; -- zx+y N:=[NC.Mul([[t]],F1), NC.Mul([[t]],F2)]; -- t*F1, t*F2 N:=Concat(N,[NC.Mul([[1],[-t]],G1), NC.Mul([[1],[-t]],G2)]); -- (1-t)*G1, (1-t)*G2 C:=[[[t,x],[-x,t]], [[t,y],[-y,t]], [[t,z],[-z,t]]]; -- set of commutators G:=Concat(N,C); Gb:=NC.GB(G,31,1,20,50); -- Done. ------------------------------- The following information printed by the ApCoCoAServer shows that Gb it is a partial Groebner basis. the number of unselected generators: 0 the number of unselected obstructions: 81 the procedure is interrupted by loop bound! the total number of obstructions: 293 the number of selected obstructions: 43 the number of obstructions detected by Criterion M: 128 the number of obstructions detected by Criterion F: 0 the number of obstructions detected by Tail Reduction: 0 the number of obstructions detected by Criterion Bk: 41 the number of redundant generators: 5 It is a partial Groebner basis.

### See also