By Peter J. Cameron
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Additional resources for Notes on Algebraic Structures [Lecture notes]
Sample text
Of course, the factorisation is not quite unique, for two reasons: (a) the multiplication is commutative, so we can change the order: 6 = 2 · 3 = 3 · 2. (b) we will see that −2 and −3 also count as “primes”, and 6 = 2 · 3 = (−2) · (−3). By convention, 1 is not a prime, since it divides everything. ) Accordingly, we will specify that irreducible elements (the analogue of primes in a general domain) should not be zero or units, and that we only try to factorise elements which are not zero or a unit.
To prove the converse, we use the Euclidean algorithm (more about this shortly), which shows that, given any two integers a and n, there are integers x and y such that xa + yn = d, where d = gcd(a, n). If d = 1, then this equation shows that xa ≡ 1 (mod n), so that xa = 1 in Z/nZ, so that a is a unit. This shows that every non-zero element of Z/nZ is either a zero-divisor or a unit. 40 CHAPTER 2. RINGS For example, for n = 12, we have: 1 2 3 4 5 6 7 8 9 10 11 unit 1·1 = 1 zero-divisor 2 · 6 = 0 zero-divisor 3 · 4 = 0 zero-divisor 4 · 3 = 0 unit 5·5 = 1 zero-divisor 6 · 2 = 0 unit 7·7 = 1 zero-divisor 8 · 3 = 0 zero-divisor 9 · 4 = 0 zero-divisor 10 · 6 = 0 unit 11 · 11 = 1 We call two elements a, b ∈ R associates if there is a unit u ∈ R such that b = ua.
B) In each case this is the “division algorithm”: we can divide a by b to obtain a quotient q and remainder r, where r is smaller than the divisor b as measured by the appropriate function d. You will have seen how to use the Euclidean algorithm to find the greatest common divisor of two integers or two polynomials. The same method works in any Euclidean domain. It goes like this. Suppose that R is a Euclidean domain, with Euclidean function d. Let a and b be any two elements of R. If b = 0, then gcd(a, b) = a.