Re: Notes on irreducibles, reducibles in Integral Domains

And does anyone know of an integral domain without
any primes?

Such domains exist in abundance: take an integral domain R with
the following properties:

1. R has only two prime ideals.
2. R is not a UFD.
3. R is noetherian.

Then R possesses no primes.

Proof: let p be a prime. Then by (1) pR is the maximal ideal of R.
Hence by (3) every non-unit of R can be written as u*p^k for some
unit u of R and natural number k. This contradicts (2).

Are there integral domains with the properties 1-3 ?
Beginner's algebraic geometry tells that the local ring in a
singular point of an algebraic curve has all of these properties.

Some commutative algebra gives the following family of examples:
take some finite extension A of the integers within a number field.
Localize A at a non-zero prime ideal; if the resulting ring R is not
integrally closed within its field of fractions, then it has the
properties 1-3.

In all of these examples none of the irreducibles is prime, but
irreducibles are at least existing. Passing to non-noetherian rings
yields examples without any irreducibles:

let S be the ring of all algebraic integers, that is complex numbers
that are roots of a polynomial x^n+a_(n-1)x^(n-1)+...+a_0
having integer coefficients.
Given a non-unit x os S, a square root y of x is in S too. Hence
x=y^2 and thus no non-unit of S is irreducible.

H
.

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