Summary. Integral Domains: Remarks and Examples
- Divisor of zero
U is a divisor of zero iff there is
V ≠ 0such that UV = 0.We found that neither Z nor Z[√m], where m is not a complete square of an integer, have no divisors of zero. But some algebraic structures do. Zm is a usual notation for the set of residues modulo m. Then, for example, in Z6, 2·3 = 0. So, in Z6, both 2 and 3 are divisors of zero.
- Function N
In the theory of numbers function N is called a norm. It is hard to overestimate the importance of this function. Function N has been used in virtually every single proof so far. Why was it needed? For
m > 0, is dense on the real line. The proof uses the Pigeonhole principle in a manner similar to that of a lemma we proved when shredding the torus. This is the reason that we had to use the norm in Euclid's algorithm. Had we used the magnitude of a number, the algorithm would have never stopped. Z[√m]
For m = -1, N(AB) = N(A)N(B), when written explicitly, takes the form
(ac - bd)2 + (ad + bc)2 = (a2 + b2)(c2 + d2),
A = a + ib, and B = c + id. This is the two-squares identity. It asserts an interesting fact: a product of the sum of two squares is itself the sum of two squares. If A = B, then N(A2) = N(A)2. On the left we have the sum of two squares. On the right - a square of an integer. This is the way to generate Pythagorean triples.
For m positive,
N(AB) = N(A)N(B)also has a nice form:
(ac + bd)2 - m(ad + bc)2 = (a2 - mb2)(c2 - md2)
This is similar to the two-squares identity. The product of two numbers in the form "a square minus a square time an integer" is an integer in the same form.
A unity (or just a unit) is a number that divides all other numbers. This is equivalent to saying that it divides 1. If U is a unity, there exists V such that
UV = 1.So that U has an inverse. The opposite is also true. Every invertible element is a unity. In a field, all elements, besides 0, are invertible. In a field division becomes ubiquitous and, therefore, not interesting.
In Z6, 2 and 3 are divisors of zero and, therefore, are not invertible. (For if
UV = 1but UW = 0with V ≠ 0,then UVW = 0·V = 0, or 1·W = W = 0.Contradiction.) On the other hand, 5·5 = 1 (mod 6).Therefore, in Z6, 5 is a unity. The remaining non-zero element 4 is a divisor of zero. For 4·3 = 12 = 0 (mod 6).Also, 4·4 = 4 (mod 6).Such elements are called idempotents. By induction, in Z6, 4n = 4.This is true for any positive n.
We may try to consider extensions of finite rings, like Zm.
Z7[√2] = Z7because, e.g., 32 = 2 (mod 7).On the other hand, if, as usual, i is taken to be a square root of -1, then Z5[i] = Z5because 22 = 4 = -1 (mod 5).