In algebraic number theory, the common definition of prime elements is this:

Definition 1. Let R be a commutative ring. An element p of R is prime if
p is not a unit and the following is true: If a and b are elements of R
such that p = ab, then a or b is a unit.

In his "Commutative Algebra", D. Eisenbud has another definition:

Definition 2. Let R be a commutative ring. An element p of R is prime if
p is not a unit and the following is true: If a and b are elements of R
such that p divides ab, then p divides a or b.

Is Definition 2 common in the commutative algebra community?

And if yes,
are there good reasons why that community uses Definition 2 instead of 1?

In article <XEmFb.1061$d%1.280494@news20.bellglobal.com>,
curious <cur@curio.us> wrote:
[snip]
In his "Commutative Algebra", D. Eisenbud has another definition:

Definition 2. Let R be a commutative ring. An element p of R is prime if
p is not a unit and the following is true: If a and b are elements of
R such that p divides ab, then p divides a or b.

Is Definition 2 common in the commutative algebra community?

It is common in algebraic number theory as well.

In article <XEmFb.1061$d%1.280494@news20.bellglobal.com>, curious
cur@curio.us> wrote:

In algebraic number theory, the common definition of prime elements
is this:

Definition 1. Let R be a commutative ring. An element p of R is
prime if p is not a unit and the following is true: If a and b are
elements of R such that p = ab, then a or b is a unit.

Actually, this is the usual definition of "irreducible", not of
prime.

[ snip ]

Do you have any reference for an algebraic number theory text that
gives Definition 1?

In article <XEmFb.1061$d%1.280494@news20.bellglobal.com>,
curious <cur@curio.us> wrote:
In algebraic number theory, the common definition of prime elements is this:

Definition 1. Let R be a commutative ring. An element p of R is prime if
p is not a unit and the following is true: If a and b are elements of R
such that p = ab, then a or b is a unit.

Actually, this is the usual definition of "irreducible", not of
prime.

In his "Commutative Algebra", D. Eisenbud has another definition:

Definition 2. Let R be a commutative ring. An element p of R is prime if
p is not a unit and the following is true: If a and b are elements of R
such that p divides ab, then p divides a or b.

Is Definition 2 common in the commutative algebra community?

It is common in algebraic number theory as well.

And if yes,
are there good reasons why that community uses Definition 2 instead of 1?

Yes, there is an excellent reason: an element p generates a prime
ideal if and only if it satisfies definition 2; but there are, in many
number fields, elements that satisfy definition 1 but do not generate
prime ideals. E.g., 2 in Z[sqrt(-5)].

I think that the "usual" definition in algebraic number theory is
the one you give as "Def. 2". Definition 1 is equivalent to Def. 2 in
a UFD, such as the integers, but it is not equivalent in most domains
that show up in algebraic number theory.

Do you have any reference for an algebraic number theory text that
gives Definition 1?

--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu

[ ... ]


As somebody else observed, both primes and irreducibles are required
to be non-zero.

Arturo Magidin wrote:

In article <XEmFb.1061$d%1.280494@news20.bellglobal.com>, curious
cur@curio.us> wrote:

In algebraic number theory, the common definition of prime elements
is this:

Definition 1. Let R be a commutative ring. An element p of R is
prime if p is not a unit and the following is true: If a and b are
elements of R such that p = ab, then a or b is a unit.


Actually, this is the usual definition of "irreducible", not of prime.

[ snip ]

Do you have any reference for an algebraic number theory text that
gives Definition 1?


Pollard and Diamond in "The theory of algebraic numbers", Chapter VII,
deal with the ring of integers in an algebraic number field K, and define:
alpha is a prime if it is not zero or a unit, and if any factorization
alpha = beta gamma into integers of K implies that either beta or gamma
is a unit.

Arturo Magidin wrote:
In article <XEmFb.1061$d%1.280494@news20.bellglobal.com>, curious
cur@curio.us> wrote:

In algebraic number theory, the common definition of prime elements
is this:

Definition 1. Let R be a commutative ring. An element p of R is
prime if p is not a unit and the following is true: If a and b are
elements of R such that p = ab, then a or b is a unit.

Actually, this is the usual definition of "irreducible", not of
prime.

[ snip ]

Do you have any reference for an algebraic number theory text that
gives Definition 1?


Pollard and Diamond in "The theory of algebraic numbers", Chapter VII,
deal with the ring of integers in an algebraic number field K, and define:
alpha is a prime if it is not zero or a unit, and if any factorization
alpha = beta gamma into integers of K implies that either beta or gamma
is a unit.

Actually, this is the usual definition of "irreducible", not of prime.

[ snip ]

Do you have any reference for an algebraic number theory text that
gives Definition 1?

And Paul Garrett's "Introduction to Abstract Algebra", page 106
http://www.math.umn.edu/~garrett/m/intro_algebra/notes.pdf

mareg@mimosa.csv.warwick.ac.uk wrote:
[ ... ]


As somebody else observed, both primes and irreducibles are required
to be non-zero.


Not in Eisenbud's "Commutative ALgebra".

Definition 1. Let R be a commutative ring. An element p of R is prime if
p is not a unit and the following is true: If a and b are elements of R
such that p = ab, then a or b is a unit.

Should there also be the condition that both a and b cannot be a unit?

Definition 1. Let R be a commutative ring. An element p of R is prime if
p is not a unit and the following is true: If a and b are elements of R
such that p = ab, then a or b is a unit.

but there are, in many number fields, elements that
[are irreducible but not prime]. E.g., 2 in Z[sqrt(-5)].

OK, great, but I am still curious --- at what point did the commutative
algebra authors (and some if not most algebraic number theory authors)
decide to call "irreducible" what we all call "prime" in high school
math, and to use the term "prime" for a different property? Does this go
back to Hilbert or even Dedekind?

This must be initially confusing for students in algebra courses, or is it?

I know the two notions are equivalent in the ring of integers,

but it
takes a bit of work to prove the equivalence (the first proof is
apparently due to Gauss, around 1800).

Arturo Magidin <magidin@math.berkeley.edu> wrote:

but there are, in many number fields, elements that [are
irreducible but not prime]. E.g., 2 in Z[sqrt(-5)].


There are simpler examples of a non-prime irreducible.

Let J < Q[x] be the ring of integer-valued polys: f(Z) < Z. In J
the irred 2|x(x+1) but neither 2|x nor 2|x+1.

Let T = R + x C[x] be the ring of polynomials in x with complex coefs
and with real constant coef.

In T, x is irred: x = fg => f or g of deg 0 so a unit

and x|(ix)^2 but not x|ix, so x isn't prime.

More generally let R < T be a ring extension with new unit u in
T\R. In R + x T[x]: x|(ux)(x/u) but neither factor.

E.g. Z + x Q[x], u = 2/3; note this ring lies between two UFDs
Z[x], Q[x] but isn't a UFD since irred x isn't prime, or,
explicitly: x x = ux x/u is a nonunique factorization.

Rings of this form are a rich source of (counter-) examples, e.g. see
[1] for a survey. Alternatively one may construct a generic
counterexample along the lines of my prior post [2].

-Bill Dubuque

[1] Muhammad Zafrullah, Various Facets of Rings between D[X] and K[X]
http://www.lohar.com/researchpdf/axbx.pdf

[2]
http://google.com/groups?threadm=y8zznru8v1d.fsf%40nestle.ai.mit.edu