ABSTRACT
In section 3 we carried over the well-known notions of divisibility from the ring ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq1305.tif"/> of integers to arbitrary integral domains. Now one of the key features of the ring ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq1306.tif"/> is that every number m ∈ ℤ \ { 0 , ± 1 } https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq1307.tif"/> possesses a prime factorization m = ± p 1 k 1 ⋯ p n k n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq1308.tif"/> which is unique up to the order of the factors.† We want to explore, in general, under what circumstances an integral domain has this unique factorization property.†† The first step is to define, for an arbitrary integral domain R, the analogue of the prime numbers in ℤ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315136554/b1e97aa6-f63a-4d22-960d-a98dde0e94f9/content/eq1309.tif"/> . Thus we want to consider those elements in R which can serve as last multiplicative building blocks. It turns out that there are two possible ways to generalize the notion of a prime number; these are given in the following definitions.
