Ring Homomorphisms & Z/(3^k)Z

 

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Rings like the Integers and Z/10Z (the ring of integers modulo 10) are ubiquitous. The slideshow above demonstrates how the rings Z/243Z, Z/81Z, Z/27Z, Z/9Z, Z/3Z, and the trivial ring are related via modding out by the prime ideal [3] . Each image represents a multiplication table where each color represents some member of a congruence class in Z/p^kZ.

Z_9
For instance, here is a multiplication table for Z/9Z written in colored numbers. The hues for Z/243Z are chosen so as to be as far apart from each other as possible and all of the colors are chosen to have full saturation values. At each step in the slideshow, elements which belong to the same coset, x + [3] , are blended together. For instance, [3] in Z/9Z is composed of the elements 0, 3, and 6. Each of their associated colors are blended, inevitably introducing bleaching in the saturation values. By the time the trivial ring is reached, all colors are equally mixed giving a single white element.

The initial colors assigned to each element are very important to how the colors blend under each homomorphism. In the slideshow below, the initial colors are reordered so that elements which belong to the same coset are similar in hue. Here instead of the colors ‘bleaching’ we see far away colors becoming more distinct until finally blending into white at the trivial ring.

 

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