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### Noether Rings

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Date: 10/01/97 at 22:01:47
From: kay gribben
Subject: Noether rings

What are Noether rings and how do they work?
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Date: 02/03/98 at 15:15:58
From: Doctor Sonya
Subject: Re: Noether rings

Dear Kay,

Sorry it took us so long to answer your question. We hope you have
found out what Noether Rings are by now, but just in case you haven't,
here is some information about them.

Noether Rings are studied in modern algebra, a subject usually first
encountered in college. If you are already taking modern algebra,
you'll understand the explanation well. If not, it may be a little
confusing.

DEFINITION: (Noetherian Ring)

A ring R is Noetherian (or it is called a noetherian ring) if for all
ascending chains of proper ideals-embedding, say:

I_1 < I_2 < I_3 <...

(For all j in the natural numbers, I_j (I sub j) are all ideals of R.)

there is a positive integer n such that I_n = I_(n+1); in other words,
every ascending chain has a maximal ideal. This condition is sometimes
called the A.C.C. (Ascending Chain Condition).

If you're studying modern algebra, the above definition should answer
your question.  If you haven't studied it, here is some background.

To understand Noether Rings (or better known as Noetherian Rings),
you need to know a few things about rings and ideals.

DEFINITION (algebraic structure)

A ring is an algebraic structure. An algebraic structure is a set of
things with some operation on them. A set is just a group of things.
For example, all the counting numbers, {1, 2, 3, 4, 5, ...} are a set.
So is a group of 16 alligators: {A1, A2, A3, A4, ... A15, A16}. An
operation is just something you can do to the things in the set that
fufills ceratin properties. Adding the natural numbers is an
operation. Leaving two alligators together and seeing which one eats
the other is another operation.  For example,  A2 * A4 = A2 means
alligator 2 and alligator 4 are put together, and the outcome is that
alligator 2 survived.  The * stands for the operation.

DEFINITION (ring)

Here is a more specific definition of a ring.

Let R be a set with two operations on it.  These two are usually
written * and +, although they may not be exactly like the familiar

R is a ring if it obeys the following rules:

1. a + b is in R for all a and b in R.
2. a + b = b + a for all a and b in R. (+ is commutative)
3. There is a zero element, denoted by 0, in R such that
a + 0 = 0 + a = a for all a in R.
4. For each element a in R, there is an inverse element of a,
denoted by -a such that a + (-a) = (-a) + a = 0.
5. + is associative, meaning a + (b + c) = (a + b) + c for all
a, b, c in R.
6. a*b in R for all a and b in R.
7. * obeys the distributive laws:
(i) a*(b+c) = a*b + a*c and (ii)(b+c)*a = b*a + c*a

In addition, a ring is called a ring with unity if it has a unity
element denoted by 1.  A unity element behaves in the following way:

1*r = r and r*1 = r for all r in R.

We use the number one to denote the generic unity element, but
remember, our set R could be anything, and so the unity might be
something other than 1.

Next, we'll look at some examples, and I hope you'll get a feel for
rings, as well as the abstract notions of addition and multiplication.

1) (Z,+,*): the set of integers equipped with the usual integer
addition and integer multiplication is a ring.

Explanation:

The set of integers refers to the collection of whole numbers:

{...,-4,-3,-2,0,1,2,3...} --- consists of all the negative integers,
the positive integers, and zero.

Let's check each rule to see whether Z obeys them:

We are discussing Z, equipped with the operations + (integer addition)
and * (integer multiplication). (For instance,
1 + 2 = 3 and 1 * 2 = 2.)

RULE 1.  a + b is in R for all a and b in R.

In this case, our R is Z. The question becomes: "When we take an
integer a and another integer b in Z, do we get another integer when
we add a to b; i.e. is a + b in Z?"

Looking at a few examples tell us that when we add whole numbers to
whole numbers, we do get whole numbers. So a + b is in Z whenever
a and b are in Z. Therefore, Rule No. 1 is satisfied.

RULE 2.  a + b = b + a for all a and b in R. (+ is commutative)

The next thing is to check if the integer addition obeys Rule 2. Look
at the example of 2 + 3. Is it equal to 3 + 2? Yes! In general, having
2 bananas for lunch and 3 bananas for dinner totals up to the same
number as having 3 bananas for lunch and 2 bananas for dinner, right?
So, this means that a + b = b + a for all a and b in Z.

RULE 3.  There is a zero element, denoted by 0 in R such that
a + 0 = 0 + a = a for all a in R.

With our specific example, the question is now: Is there an integer
such that no matter what you add it to, you just get back the number
you added it to?  This should sound familiar. It's just the usual 0,
because 0 + a = a + 0 = a for all a in Z.

RULE 4.  For each element a in R, there is an inverse element of a,
denoted by -a such that a + (-a) = (-a) + a = 0.

Is that true for Z?  Let's see...

Take an integer say 4.  What corresponding element would, when added
to 4, turn the sum to zero?  Well, -4 of course!

In general, given an integer a, the corresponding integer -1*a (often
conveniently denoted by -a) will be such that a + (-a) = (-a) + a = 0.

So Z satisfies the above rule.

RULE 5.  + is associative, meaning a + (b + c) = (a + b) + c for all
a, b, c in R.

The question is: "Is +, or integer addition, associative?"

Look at 2 + 3 + 4. It is equal to both (2+3) + 4 as well as 2 + (3+4),
right? In fact, in general, (a + b) + c = a + (b + c) for all a, b, c
in Z.

RULE 6.  a*b in R for all a and b in R.

This is the first rule that involves the multiplication *.

Look at the integer multiplication in Z.

Is the product of two integers an integer?

Yes! The definition of integer multiplication gives us evidence. Look
at the meaning of integer multiplication:

2 x 3 means the number of objects found in an array of objects
arranged in a 2 by 3 rectangle:

*  *  *
*  *  *

totalling up to 6 objects. No matter which two integers you multiply
together, you'll get a rectangle like this, and it will have an
integer number of entries. So the answer to a multiplication will also
be an integer. In fact, for any a and b in Z, a*b = a x b is an
integer, thus a member of the set Z.  So, Rule 6 is satisfied.

RULE 7.  * and + obey the distributive laws:
(i) a*(b+c) = a*b + a*c
and (ii)(b+c)*a = b*a + c*a

Let's check to see whether (Z,+,*) obeys these distributive laws.

Take three integers, say 4, 5, 6.

4*(5+6) = 4 x (5 + 6) = 4 x 11 = 44

4*5 + 4*6 = 20 + 24 = 44

So, this hints that Z obeys the distributive law (i). ("hints" because
we haven't proven that it is true for all elements in Z).

Similarly, (ii) seems true since:

(5+6)*4 = 11 x 4 = 44
5*4 + 6*4 = 20 + 24 = 44

In fact, both (i) and (ii) are true.  Why?

We can show you pictorially why this is so. To prove the general case,
look at the following arrays of stars:

The first two arrays represent 2 x 3 and 2 x 4 respectively:

*  *  *         *  *  *  *
*  *  *         *  *  *  *

When we add them up, we join the two arrays together:

*  *  *  *  *  *  *
*  *  *  *  *  *  *

In the end, we have two rows of 7. Where did we get the seven?  We got
7 because we joined up 3 columns and 4 columns!

So, in general, can you use a similar demonstration to show why:

a (b + c) = ab + ac and (b + c) a = ba + ca  ?

Whew! We've checked all of the rules, and our sample ring follows all
of them, so we can conclude that (Z,+,*), the set of integers Z
equipped with the usual integer addition + and the integer
multiplication *, is a ring.

Note that the ring operations + and * do not always mean the usual
addition and multiplication we are used to! These + and * operations
are binary operations, meaning that they take two objects from the set
R, and somehow combine and act on these two objects to produce another
object in the set. For instance, if the underlying set is the set of
all colours and + is the operation of mixing the two colours to
produce another colour, then the following is true from our experience
with colours:

Blue + Yellow = Green, and
Yellow + Red = Orange.

Green and Orange are still colours, and in this case + means "Mix 'em
together."

Indeed, Yellow + Blue = Green and Red + Yellow = Orange. So, mixing
colours (in painting, not primary light colours) is a commutative
operation.

Of course, we can talk about a zero element in the set of colours. We
know that we can produce paint with no colour (colourless paint). So,
by mixing colourless paint with red paint (provided the colourless
paint is chemically inactive with our other paints), we still get red
paint as a result!

In painting pictures, we can accidentally paint the wrong colours or
spill paint over the paper. Imagine that the factory comes up with the
following set of paints: To make red paint colorless, the factory has
manufactured a special colour-paint called "der" so that when we apply
this "der" paint to the red paint, we obtain (via a chemical reaction)
the colourless paint we mentioned earlier. To make blue colourless, we
need to apply another "eulb" paint to the blue paint. For each colour,
there is a certain "inverse" paint that helps it revert to a
colourless paint. In this case, our operation + on the set of all
colours satisfies Rule 4.

Rule 5 is also satisfied, since:

Blue + (Red + Green) = (Blue + Red) + Green.

So, +, colour mixing, is an example of an addition operation that can
be defined on the set of all colours.

To summarise, both + and * are operations that are abstractions of our
usual experience of addition and multiplication. The properties/rules
they satisfy, together with the underlying set R, make up the whole
ring.

Here are some other examples of rings; you may use the rules to verify
that they are indeed rings:

2)  (Z[X],+,*): the set of polynomials in X with integer coefficients,
equipped with the usual addition of polynomials and usual
multiplication of polynomials is also a ring.

3)  (M2(R),+,*):  the set of 2 by 2 real matrices is also a ring,
equipped with the usual matrix addition and matrix multiplication.

Next we'll look at the first example (Z,+,*) again, and define some
more terms.

Take a look at the subset of all even integers, denoted by 2Z. The
elements are:

...-6, -4, -2, 0, 2, 4, 6 ...

Say we take any element from Z, say k, and we multiply k on the left
of all the elements in 2Z. What do you observe?

Probably you can see that  ..., -6k, -4k, -2k, 0, 2k, 4k, 6k are all
elements of 2Z (because they still even integers).

Morevover, any two elements in 2Z still sum up to be in 2Z; ie, if
j and k are in Z, then

2k + 2j = 2(k+j)

still in 2Z (because k+j is in Z).

So we find that 2Z is a special subset in Z which satisfies the
following properties:

(a)  a + b is in 2Z for all a, b in 2Z.
(b)  ka is in 2Z for all k in Z and all a in 2Z.

If we generalise this notion for any ring, we have the following
definition of an ideal:

DEFINITION (Ideal of a ring)

Let (R,+,*) be a ring. A subset I of R is called an ideal of R,
denoted by I<R if the following criteria are satisfied:

(i)  i + j is in I for all i,j are in I.
(ii)  r*i is in I for all r in R and i in I.

Examples of ideals:

(i)  mZ which is defined to be the set {mk | k in Z} (i.e. all the
integer multiples of m, is an ideal of (Z,+,*).

(ii)  The set {Xf(X) | f(X) in Z[X]} is an ideal of (Z[X],+,*).

Now, if you look at the integer ring (Z,+,.) again, you will discover
that ideals can be found embedded (or contained) in other ideals. For
instance, all multiples of 6 are found in the multiples of 3.

The interesting thing is that 3Z cannot be found embedded in another
ideal (not equal to Z istelf)! We'll sketch a proof of this theorem
for you, but you'll have to fill in some of the details yourself:

THEOREM:  3Z cannot be found embedded within another ideal.

PROOF:  Now all the ideals of Z must be of the form mZ, for some
integer m.

Can you figure out why? Suppose that there is a proper ideal, say pZ,
that contains 3Z and is not equal to it.

A proper ideal is just an ideal that is not the whole ring. Check to
see that the entire ring is also an ideal, albeit a boring one.

3 is in 3Z, and since 3Z is in pZ, 3 is also in pZ.

This means that pk = 3 for some k in Z.

But wait, 3 is a prime number! Thus its only factors are 3 and 1, so
pk must equal 3*1 or 1*3.

This leads us to conclude that either p is 1 or k is 1.

But if p is 1, then pZ is the whole ring (do you see why?) and that
can't be so.  Thus k must be 1 and p must be 3.

If p is 3, then pZ = 3Z, but we also don't want this to be true.

So p can't be anything, and so the ideal pZ that contains 3Z but is
not equal to R can't exist.

We thus conclude that 3Z cannot be embedded into another proper ideal
in Z.

Now we are led to the next (and last) definition.

DDEFINITION (Maximal Ideal)

I is a maximal ideal of R if I cannot be embedded into another proper
ideal of R.

For example, pZ is a maximal ideal of Z for all p primes in Z. Can you
prove why this is true?

Finally, we can come to the definition of a Noether Ring (Noetherian
Ring).

DEFINITION (Noetherian Ring)

A ring R is Noetherian (or it is called a noetherian ring) if for all
ascending chains of proper ideals-embedding, say:

I_1 < I_2 < I_3 <...

(For all natural numbers j, I_j (I sub j) are ideals of R.)

there is a positive integer n such that I_n = I_(n+1); in other words,
every ascending chain has a maximal ideal. This condition is sometimes
called the A.C.C. (Ascending Chain Condition).

For example, (Z,+,*) is a Noetherian Ring. This is because any
ascending chain of ideals must terminate at the ideal pZ for some
prime number p.

Feel free to write us back if you have any more questions.

-Doctord Sonya and Joe,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
College Definitions
College Modern Algebra

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