Re: Algebra with quotient ring.

Tonico <Tonicopm@xxxxxxxxx> wrote:
Bill Dubuque <wgd@xxxxxxxxxxxxxxxxxxxx> wrote:
mina_world <mina_world@xxxxxxxxxxx> wrote:

A commutative ring A is called a principal ideal ring
if every ideal of A is principal.

Show that a quotient ring of a principal ideal ring
is principal ideal ring.

In fact, I want to know that Z_n is principal ideal ring.
so, If I can show above problem, this is trivial by Z/(nZ).
But... I can't. so, I need your advice.

See my prior post
http://google.com/groups?selm=y8z8yfyyhsn.fsf%40nestle.csail.mit.edu

Bill, you mislead with your (in)famous "see my prior post" thing:
the link you gave talks about localizations, not quotient rings...*sigh*.

Apparently you don't know enough algebra to see that the
same idea in my linked post applies whichever denotation
of "quotient ring" the OP intended, i.e. whether it
denotes a localization or a factor/residue ring. Both
denotations are in widespread use in abstract algebra.

This misleads and confuses sombody wanting something
pretty basic and elementary

I don't compose my posts only for the OP - they are read
by a much wider audience, both now and in the future (e.g.
in archive searches). So even if (part of) my post was
actually over the OP's head (which there is usually no
way to know in general a priori) it may still prove quite
helpful to many other readers of the post.

not to mention your unbased claims like

The fraction ring construction D S^-1 with denominator
submonoid S < D* preserves many properties of domains D,
e.g. Euclidean, PID, UFD, valuation, Bezout, GCD, Dedekind,
Prufer, Krull, Noetherian, integrally closed, etc

...no proofs, no links...and that "etc." one might really be
frightening to someone asking such an easy question as the OP.
A pitry, really...

Again you win the prize for most bizarre response to an
interesting post. By your logic one should never mention
anything beyond a students knowledge because it might be
so "frightening". Oh the horror that the student might
actually be motivated to investigate some interesting
generalizations and thereby acquire a deeper understanding
of the subject under discussion. In any case, in the part
of my post that you snipped is a sketch of an elementary
proof of said property of localizations - nothing very
"frightening" at all. Your remark that "etc" might be
"really frightening" will surely make it into my log of
most ridiculous remarks I've encountered. Thanks so much
for the laugh. That's almost as hilarious as your prior
remarks that were as naive as the satirical review of
Dummit & Foote, and your reply that took it seriously.

A pitry, really...

Really, I did not try pi in any of those exotic domains.
Is it fruitful? Should I try a cut of Dedekind? Any eta?

Why do you call my remarks "unbased claims"? If you think
they're in error then do explain why. I do not see such.
Of course there are "no proofs". The remarks were an aside
for cultural enrichment. You'll find similar remarks in
many well-written textbooks and expository papers. One
can easily Google the terms if one desires to learn more.

Do you really expect me to post links for you after you
so rudely complained at length before about how difficult
it was for you to follow all the links in my posts. Sorry,
I'm not your hired tutor. If you can't be bothered to click
your mouse a few times to follow a link then you have not
demonstrated your sincerity to learn mathematics. If it
hurts your brain to navigate a tiny graph of linked posts
then you'll have a very difficult time learning much higher
mathematics - whose many beautiful subjects are linked in
many complex, mysterious and tantalizingly beautiful ways.
One of the goals of my posts is to provide links to many
of these beautiful mathematical interconnections. This is
something one doesn't normally find in textbooks or in
by-the-book replies in other run-of-the-mill followups.

This misleads and confuses...

If all those big mathematical words confuse you then simply
ignore them. Better, go back to your favorite pastime here
- namely bashing cranks (which accounts for a significant
percentage of your sci.math posts, second only to Uncle Al).
There you should find comfort in the fact that the cranks
won't confuse you so much with many big mathematical words.
And there you might actually comprehend the cranks' jokes,
instead of being the *** of them as here.

--Bill Dubuque
.

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