Noether RingsDate: 10/01/97 at 22:01:47 From: kay gribben Subject: Noether rings What are Noether rings and how do they work? 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. Here's the short answer to your question, "What are Noether Rings?": 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 addition and multiplication you know. 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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