"Abstract algebra" 05-Rings and Fields

Abstract algebra is not abstract, but the algebraic system studied has a wide range of example prototypes. In the study of group theory, we have seen that many systems have two operations at the same time, and they are interrelated, which forces us to study the structure and characteristics of this algebraic system. On the other hand, the interaction between operations can also lead to the special nature of a single operation, which you will see later in the discussion.

1. Ring 1.1 ring and sub-ring

The system with two operations is more and the nature is different, we must first extract the "smallest" system to be universal. The addition and multiplication of various numbers, polynomial, and matrix are the most representative dual-operation systems, and they can be used as reference to obtain more useful systems. The Matrix (linear space) provides a wealth of possibilities for a dual-operation system, and the examples in the textbook can be avoided, but you only need to know some basic concepts, and linear algebra will be discussed in the future as a special topic.

Examining the common systems mentioned above, their addition groups are commutative groups, so it is assumed that an operation of the new abstract system is also a commutative group. For convenience, it is directly called Dabigatran , and the unit element of the group is called the 0 element (recorded as \ (0\)). All expressions of Dabigatran can be written as addition and subtraction, and the "power" of addition can be expressed in multiples. You can prove that the following common variants are established and can be used directly in the future.

\[a+0=a;\quad A-a=0;\quad-(-a) =a;\quad-(A+b) =-a-b;\quad-(A-B) =b-a\tag{1}\]

\[-(NA) = (-N) a;\quad ma+na= (m+n) a;\quad m (NA) = (MN) a;\quad n (a+b) =na+nb\tag{2}\]

The multiplication group in the above system is weaker, but at least the combination law is established, so it is a semigroup. If we define a system with only two isolated operations, it is not necessary to do such a study. In the study of common systems, multiplication and addition can be found to meet the following distributive laws . Now we can define a new system, one operation is an addition group, the other is a semigroup, and they satisfy the distribution rate, such a system is called the ring , which is generally denoted by the letter \ (r\), and the commutative ring is called the commutative ring . Multiplication, if there is a unit element, is generally written in accordance with the Convention \ (1\).

\[a (b+c) =ab+bc;\quad (a+b) c=ac+bc\tag{3}\]

If you look closely at the distribution rate, you can see that there is a shadow of the homomorphism mapping, which is also the main reason for all kinds of properties. Now let's look at the nature of addition in the combination of multiplication, and the expressions that we used to be familiar with can still be formed. First for the special \ (0\) element, because \ (0a+0a= (0+0) a\), easy to have \ (0a=0\), 0 elements under multiplication will all the elements classified as \ (0\). Look again \ ((-a) b\), because \ ((-a) b+ab= (-a+a) b=0\), so ((a) b=-ab\). These are expressions that were familiar to the past, and they are all set up in the ring. Here are more common expressions that prove easier.

\[0a=a0=0;\quad (-a) b=a (-B) =-ab;\quad C (a) =ca-cb\tag{4}\]

\[\sum{a_i}\sum{b_j}=\sum\sum{a_ib_j};\quad (MA) (NB) = (MN) (AB) \tag{5}\]

Naturally, it is possible to define a sub-ring , which is a basic definition for further study of the ring structure. In addition to being a subgroup of Dabigatran, the sub-rings need to be closed for multiplication, which is more easily demonstrated. Like a single-op system, a ring isomorphism can be defined, if there is a one by one mapping \ (f\) between two rings \ (r_1,r_2\), and the map hold operation (equation (6)) is called \ (r_1,r_2\) isomorphic (\ (R_1\cong r_2\)).

\[f (a+b) =f (a) +f (b); \quad f (AB) =f (a) f (b) \tag{6}\]

In the ring, we can also discuss the unit element and inverse, because of the interaction of the two operations, it tends to show interesting properties, but the proofs also need ingenious constructs. For example, to investigate a unit of the ring \ (r\), if the element \ (a\) has at least two right inverse, the set of the inverse meta is \ (\{a_k\}\). It is easy to prove that they are all right inverses and are not equal, so that the two sets have a one by one mapping. \{a_ka-1+a_1\}\ If the collection is finite, there is \ (a_xa-1+a_1=a_1\), and the two sides are multiplied by \ (a_y\) (a_x=a_y\), so that all right inverses are equal. This leads to contradictions, so there must be an infinite number of right inverse, and the finite ring can only have a right inverse, this conclusion is called Kaplanskey theorem.

　　• Each element is an idempotent (a^2=a\) ring called a Boolean ring, proving \ (a=-a\) and \ (ab=ba\);

　　• the ring \ (r\) has a unit element, proving that the addition exchange law can be proved by other parts of the definition; (Hint: non-commutative distribution rate)

　　• Verification: The only left (right) unit element must be the unit element; (Hint: construction \ (ae_l-a+e_l\))

　　• if \ (1+ab\) is reversible, then \ (1+ba\) is also reversible; (Hint: construct)

　　• Verification: All elements in the swap ring satisfy \ (a^n=0\) constitute a sub-ring.

1.2 0 factors

The 0 element has a special position in the ring, which is like a black hole in which all elements are inhaled, causing the partial appearance of the ring to collapse. Conversely, the overall structure of the ring depends on those elements that can escape the "Gravity" of the 0\ to brace up. For this definition \ (ab=0\) \ (A,b\ne 0\) is the left and right 0 factors of the ring respectively, the element that is not the 0 factor is called the regular element, and the regular element is the "supporting element" we are looking for. No zero factor is a constraint on multiplication, which essentially requires the multiplication to be closed. There is a special kind of 0-factor satisfying \ (a^n=0\), which is called a power of 0 yuan .

Obviously there is no left 0 factor and there is no right 0 factor is equivalent, such a ring is also called zero-factor ring , the exchange of zero-factor ring called the whole ring (domain) (some textbooks also require unit element, not used here). For zero-factor loops, if there is a \ (ab=ac\) or \ (ba=ca\), the distribution rate obviously has \ (b=c\), that is, the elimination Law is established. Conversely, if the left (right) elimination law is established, it is also easy to obtain a ring zero factor, thus eliminating the law and zero-factor is two equivalent concept.

If the zero-factor ring has left unit element \ (e_l\), because \ ((ae_l-a) e_l=0\), then there is \ (ae_l=a\), so the ring has a unit element, with the same method can be proven its left inverse is also the right inverse element. This example shows the rich effect of the 0-factor concept on the establishment of an equation, and is adept at constructing ingenious expressions that can be used to get a lot of useful conclusions. Some scenarios may not exist in the unit element, to \ (ab=a\) can not rush away \ (a\), but to bypass the use of elimination law, this method is often used. For example, if no zero factor ring has idempotent (x^2=x\), can not directly eliminate to get \ (x=e\), but first multiply on any element \ (ax^2=ax\), and then eliminate the \ (x\) proof \ (x\) is the unit element. Consider the following questions:

　　 if \ (S\leqslant r\), but their unit element is different, verification \ (s\) unit element is \ (r\) 0 factor;

　　• A Boolean ring containing at least \ (3\) elements is not an integral ring;

　　• if there is \ (ab=1\) in the finite ring, then \ (ba=1\). (Hint: reference to the proof of Kaplanskey theorem)

1.3 Characteristics

The order is an important parameter of the group, now look at the order of the elements in the addition group, if there is a maximum value \ (n\), because the addition group is a commutative group, with contradiction know all elements of the order is \ (n\) factor. The Order of the addition group has more properties in the ring, we will call the largest order of the ring characteristics , remember as \ (\text{char}\:r\), of course, the characteristics can also be infinite. If the multiplication has a unit element and the order is \ (n\), then there is \ (na= (n1) a=0\), so the order of \ (1\) is the characteristic of the ring. The characteristic is \ (p\), and the constant (a^p=a\) ring called \ (p\)-ring, it can be proved that \ (P\)-ring is a cyclic ring (more complex).

The multiplication in the ring has a very useful property, that is, the multiples can be arbitrarily moved together (equation (5)), this feature combined with zero-factor can be very good properties. Assuming that there is a n\ element in the ring (a\), then according to the \ ((NA) b=a (NB) =0\), it is easy to know the factor of \ (b\) as the Order \ (n\), and then know that the order of all elements in the ring is \ (n\). Assuming that \ (n\) is not a prime number, it has decomposition \ (n=xy\), then there is a \ ((NA) a= (xa) (ya) =0\), so there must be \ (xa=0\) or \ (ya=0\), this and \ (A\) the Order of \ (n\) contradiction. The above analysis shows that the order of no-zero factor ring element is either infinite or a prime number \ (P\), and the order of the Finite zero factor ring is of course \ (p\).

　　• If the swap ring is characterized by \ (p\), then there is \ ((\sum{a_k}) ^p=\sum{a_k^p}\);

　　• verification: \ (p\)-the ring does not have a power of 0 yuan.

2. Except ring and domain 2.1 In addition to rings and domains

Some rings have more properties on multiplication and it is necessary to specifically discuss them. For those with a unit element, the element that exists in the inverse is generally called the unit . It is easy to prove that the whole unit in the ring forms a group under multiplication, which is called the unit group . For the finite ring, there is always \ (A^m=a^n (m>n\geqslant 1) \) is established, if \ (a\) is a non-0 factor, then there is \ (a^{m-n+1}=a\). then to any \ (x\) has \ (a^{m-n}x=x\), that is, to get \ (a^{m-n}\) is the unit, and \ (a^{m-n+1}\) is the inverse of \ (a\). Summing up the above, the non-0 factor of the finite ring is the unit.

In addition to the 0 factors, each element is a unit of a ring called a ring or body (skew field), and the commutative ring is also called the field (field). It is easy to prove that there are no 0 factors in the ring except that the multiplication is still closed after removing the 0 elements, which can form a group. The number system is a typical representation of the ring and the domain, the integer ring has the unit \ (\{-1,1\}\), the rational number, the real, the plural are the domain examples. Because domain multiplication can be exchanged, you can define \ (ab^{-1}=b^{-1}a\) as fractional \ (\dfrac{a}{b}\), you can prove that the general way of addition, multiplication, in addition to the law in the domain is also established.

　　• if the ring \ (r\) in any non-0 elements \ (a\), there is a unique \ (b\in r\) make \ (aba=a\), Verification \ (r\) for the removal of the ring. (Hint: no zero factor in the first pass)

You may have a question about the existence of a ring in addition to it? Does multiplication have a unit element and an inverse, but not commutative ring exists? Remember the first chapter introduced in the four-tuple, by them as "super complex" units formed four yuan (\{a+bi+cj+dk\}\), can prove that it is a non-commutative de-ring. This is the first non-commutative removal ring found in history, first discovered by Hamilton (Hamilton), and therefore also called Hamilton four Yuan in addition to the ring. Later in the course, it is also described as the general nature of the number of satisfied, which is historically a significant discovery. For the finite de-Weidebong (Wedderbum), it is proved that it must be exchanged, so it must be a domain. From the previous discussion we are easy to have, the finite zero-factor ring is bound to be in addition to the ring, and then by the Weidebong theorem know it and must be a domain.

Hamilton (1805-1865)

In the definition of the previous group, we discuss the equivalence between the solution of the first equation and the group. Similar conclusions are found in the ring, and the requirements are weaker. The first equation (7) In addition to the ring has a solution, and conversely if the ring satisfies the equation (7) One of which has a solution, the following is whether it is in addition to the ring. First, there is no zero factor, that is, the arbitrary \ (A,b\ne 0\), proof \ (Ab\ne 0\). You can construct an expression containing \ (ab\) and a value of \ (b\) (or \ (a\)), with the solution of the first equation can have \ (bc=d\) and \ (ad=b\), thus \ (abc=ad=b\), then the Ring zero factor. Next find the unit element, set \ (ax=a\) the solution for \ (e_r\), using the elimination law (see above discussion) (e_r\) is the right unit element. Again by \ (ax=e_r\) know any element \ (a\) has a right inverse, thus multiplication (minus 0 yuan) is a group, the ring is the other ring. In the above discussion, the necessary and sufficient condition of the ring for the removal of the ring is a constant solution (\ (A\ne 0\)) of one equation (7).

\[ax=b,\quad Ya=b\tag{7}\]

2.2 Business Domain

The structure of the domain is the most common, its conclusion is rich, we want to be able to put a ring in the domain, in order to get more conclusions. Obviously not all rings can be extended to a domain, it must satisfy at least zero-factor and exchangeable. Naturally, we would like to ask, is not the first consideration of zero-factor non-commutative ring extension to the ring, unfortunately this conclusion has been cited counter-examples, more complex, here only when the conclusion. So is no zero factor commutative ring (whole ring) can be extended to the domain? This is the question to be discussed here.

To be a domain, you need to replenish the unit and inverse, but it's hard to define them. Looking back at the extension method we introduced in the real-number system, we can define the extended number of systems with several pairs, and then embed the original numbers into the new number series. Adding the unit element and the inverse essentially requires division, and the integer extension is exactly the same as the process of the rational number, defining the set of element pairs \ (\{(a, b) \} (a,b\in R, A\ne 0) \). When \ (ad=bc\), the definition of equality \ ((b) = (c,d) \), intuitively speaking, is actually defined as the fractional \ (\dfrac{b}{a}=\dfrac{d}{c}\).

Equivalence classes under equal relations are exactly what we expect, first proving that the following additions and multiplication definitions of the new system are benign, that is, the selection of representative elements in the equivalence class does not affect the results. It then proves that the new system forms a domain under this operation definition and finally embeds the ring into the domain by mapping \ (A\to \dfrac{ad}{d}\). This proves that no 0 swap ring can always be extended to a domain, which is also called the fractional domain or quotient domain of the ring.

\[\dfrac{b}{a}+\dfrac{d}{c}=\dfrac{bc+ad}{ac},\quad \dfrac{b}{a}\cdot\dfrac{d}{c}=\dfrac{bd}{ac}\tag{8}\]

3. Special Ring 3.1 cycle ring

The circle group is the simplest group, and the first analysis is that the addition group is the loop of the cyclic group, which is called the cyclic ring , and the generating element of the addition group is \ (a\). Looking back at the cyclic group, if its order is infinite, it is isomorphic to the integer group and the same as \ (-a\) is the generating element, Jorgia is finite \ (n\), it is isomorphic to the remainder class group of \ (n\), and any remaining class with the \ (n\) is a generating element. It is obvious that the set of integers \ (z\) and any remaining class sets \ (Z_n\) Form rings under addition and multiplication definitions, respectively, as integral rings and modulo \ (n\) remaining class rings , and then analyze the loop and the relationship between them.

First look at the loop loop, all of its elements are \ (\{\cdots,-2a,-a,0,a,2a,\cdots\}\) or \ (0,a,2a,\cdots, (n-1) a\). They have only one homogeneous cyclic group in each of the different orders under the addition group, but this is not true in the ring. Now consider the multiplication in the loop loop, first of all two elements known as \ ((MA) (na) = (na) (MA) = (MN) a^2\), so the loop must be a commutative ring. Secondly, by the closure of the multiplication, there must be \ (a^2=ka\), and conversely if the multiplication is defined on a cyclic group ((MA) (NA) = (MNK) a\), it must also constitute a ring. It can be concluded that any value of \ (k\) is equivalent to a ring structure, and of course you have to understand that different \ (k\) corresponding rings are likely isomorphic.

For an infinite order ring, the addition generator is only \ (\pm a\), when \ (|k|\) takes different values, the corresponding rings must not be isomorphic, and it is easy to prove that the corresponding rings of \ (k\) and \ (-k\) are isomorphic. For the \ (n\) Order ring, \ (k\) can only take \ (n\) values, and the corresponding rings of these values may be isomorphic. Using some simple derivation of elementary number theory, it is easy to prove that a factor of \ (k\) to \ (n\) can be made by selecting the appropriate generating element. Thus, each factor of \ (n\) represents a class of homogeneous rings, and the rings corresponding to different factors are different in structure.

So that all the homogeneous rings of the loop are clear, each nonnegative integer corresponds to an infinite ring, and each factor corresponds to a \ (n\) Order ring. Finally, we look at the integer loops and the remaining class rings, and obviously their generators meet \ (k=1\), and their sub-rings satisfy \ (k>1\). All of \ (z\) are corresponding to the positive integer one by one, and all the sub-rings of \ (z_n\) correspond to the positive factor one by one of \ (n\), and they contain all loop loops except \ (k=0\). In other words except \ (k=0\), each loop is isomorphic to a sub-ring of \ (z\) or \ (z_n\).

Now to do some general discussion, \ (z\) only the invertible element \ (\pm 1\), all elements are non-0 factors, \ (z_n\) and \ (n\) are reversible elements, and the others are 0 factors. In particular, each element of \ (z_p\) is reversible, so it is a domain, and also a \ (p\)-ring. Since \ (z\) and \ (z_n\) have unit elements, the order of the units is their characteristics, so \ (z\) is characterized by infinity, and \ (\text{char}\: z_n=n\).

3.2 Polynomial ring

Extending the loop to multidimensional space is a common method for obtaining more complex rings, and the forms of expansion are also varied. The matrix ring can get very rich ring structure, simple and simple expansion in linear space, such as irrational number ring \ (\{x+y\sqrt{2}\mid x,y\in\bbb{q}\}\) and complex ring \ (\{x+yi\mid x,y\in\bbb{r}\}\), Specifically, \ (\{x+yi\mid x,y\in\bbb{z}\}\) is called the Gaussian integral ring .

The most common polynomial in linear extension, polynomial is always an important concept in algebra, it is a basic algebraic object, now analyze polynomial system from the angle of ring. Starting with the most common unary polynomial, it is an expression with the following form, where \ (a_k\) is the element of the ring \ (r\), \ (a_kx^k\) is called \ (k\), \ (a_k\) is called the \ (k\) sub-coefficient, the maximum number of coefficients nonzero is called the number of times of the polynomial.

\[f (x) =a_nx^n+\cdots+a_1x+a_0\tag{9}\]

We are initially aware of the polynomial in the context of the domain, where the limitations require a redefinition of the general understanding of the polynomial. First the plus sign in the polynomial and the "multiplication sign" in \ (A_kx^k\) is currently only a tick, and the necessary and sufficient condition for the equality of two polynomial is that the coefficients of each item are equal, not the final value. Now the operation needs to be redefined, the sum of the two identical entries \ (a_kx^k,b_kx^k\) is \ ((a_k+b_k) x^k\), and the product of the two items \ (a_ix^i,b_jx^j\) is \ (a_ib_jx^{i+j}\), Two polynomial multiplication is carried out by the allocation rate. The above definitions do not need to be defined for rings on a domain, but must have such an exact description under the ring.

It is easy to prove that the polynomial set on the ring \ (r\) forms the ring under the definition of addition and multiplication above, and is generally written as \ (r[x]\). Obviously \ (r\) is the sub-ring of \ (r[x]\), so the general set of \ (r[x]\) (r\) also must be set up, but the reverse conclusion is generally to prove. There are some obvious conclusions, such as if \ (r\) has a unit element (r[x]\) also has a unit element, if \ (r\) can be exchanged (r[x]\) can also be exchanged, if \ (r\) for the whole ring then \ (r[x]\) is also an integral ring. Now look at the nature of \ (r[x]\) 's 0 factor, assuming \ (f (x) g (x) =0\), set \ (g (x) \) The first coefficient is \ (g_n\), then \ (f (x) g_n\) is smaller than \ (f (x) \). If it is assumed that the ring can be exchanged, then there is a \ ((G_NF (x)) g (x) =0\), which is known to exist by the inductive method (C\in r\), so that the \ (CG (x) =0\). To summarize the above discussion there is a necessary and sufficient condition that the commutative ring \ (r[x]\) element \ (g (x) \) is a 0 factor, exists \ (CG (x) =0\).

The decomposition of Integer ring (arithmetic basic theorem) is an important content of elementary number theory, which can still be discussed in the general circle, and the topic will be given later. In addition to the polynomial ring, there is also an important Gaussian integer, which is also an important ring. To extend the polynomial into a domain, the rational fractional domain must be introduced.

"Abstract algebra" 05-Rings and Fields