Editing Ideal (ring theory) (section)

== Examples of ideal operations ==
In <math>\mathbb{Z}</math> we have
: <math>(n)\cap(m) = \operatorname{lcm}(n,m)\mathbb{Z}</math>
since <math>(n)\cap(m)</math> is the set of integers that are divisible by both <math>n</math> and {{tmath|1= m }}.

Let <math>R = \mathbb{C}[x,y,z,w]</math> and let {{tmath|1= \mathfrak{a} = (z, w), \mathfrak{b} = (x+z,y+w),\mathfrak{c} = (x+z, w) }}. Then,
* <math> \mathfrak{a} + \mathfrak{b} = (z,w, x+z, y+w) = (x, y, z, w)</math> and <math>\mathfrak{a} + \mathfrak{c} = (z, w, x)</math>
* <math>\mathfrak{a}\mathfrak{b} = (z(x + z), z(y + w), w(x + z), w(y + w))= (z^2 + xz, zy + wz, wx + wz, wy + w^2)</math>
* <math>\mathfrak{a}\mathfrak{c} = (xz + z^2, zw, xw + zw, w^2)</math>
* <math>\mathfrak{a} \cap \mathfrak{b} = \mathfrak{a}\mathfrak{b}</math> while <math>\mathfrak{a} \cap \mathfrak{c} = (w, xz + z^2) \neq \mathfrak{a}\mathfrak{c}</math>
In the first computation, we see the general pattern for taking the sum of two finitely generated ideals, it is the ideal generated by the union of their generators. In the last three we observe that products and intersections agree whenever the two ideals intersect in the zero ideal. These computations can be checked using [[Macaulay2]].<ref>{{Cite web|url=http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_ideals.html|title=ideals|website=www.math.uiuc.edu|access-date=2017-01-14|archive-url=https://web.archive.org/web/20170116190119/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_ideals.html|archive-date=2017-01-16|url-status=dead}}</ref><ref>{{Cite web|url=http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_sums_cm_spproducts_cm_spand_sppowers_spof_spideals.html|title=sums, products, and powers of ideals|website=www.math.uiuc.edu|access-date=2017-01-14|archive-url=https://web.archive.org/web/20170116185903/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_sums_cm_spproducts_cm_spand_sppowers_spof_spideals.html|archive-date=2017-01-16|url-status=dead}}</ref><ref>{{Cite web|url=http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_intersection_spof_spideals.html|title=intersection of ideals|website=www.math.uiuc.edu|access-date=2017-01-14|archive-url=https://web.archive.org/web/20170116185829/http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.9.2/share/doc/Macaulay2/Macaulay2Doc/html/_intersection_spof_spideals.html|archive-date=2017-01-16|url-status=dead}}</ref>