Re: Algebraic questions

In article <4dcb50e8-a454-4cca-8e7d-c65a279f5ee7@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
Tonico <Tonicopm@xxxxxxxxx> wrote:
On May 2, 3:33=A0pm, Mariano Su=E1rez-Alvarez
<mariano.suarezalva...@xxxxxxxxx> wrote:
On May 2, 9:02 am, sanchopanch...@xxxxxx wrote:

Hello,

suppose that (f_1,...,f_n) is a prime ideal in a commutative unitary
ring and f_1,...,f_n a minimal system of generators. Is it true that
f_1,...,f_n are prime elements in this ring? I suppose not but can't
even construct a counterexample.

The ideal (6, 10) is prime in Z, and the given set
of generators is clearly minimal.

-- m

"*******************************************************

Wasn't Z a PID? Then how come can there be an ideal with a minimal set
of generators of size two??

There is a problem of nomenclature here.

You can (partially) order generating sets by inclusion; in that case,
a generating set X is "minimal" if and only if for every proper subset
Y of X, the ideal (Y) is properly contained in the ideal (X).

Or you can compare generating sets by size/cardinality. This is a
pre-order, but one we are very comfortable with; the generating set X
would be "minimal" under this preordering if and only if for every
generating set Y, if |Y|<=|X| then |Y|=|X|.

The set {6,10} is a minimal generating set for the ideal (2) in Z
under the first meaning, but not the second.

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

Arturo Magidin
magidin-at-member-ams-org

.

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