fsb4000 wrote:

2. a) a divisible abelian group has no maximal subgroups

I think that's enough complete solutions, right? After all, the moderators will bury me for the fact that I have already completely painted two tasks for you !!! Therefore, in order not to anger them, we will limit ourselves to ideas.

Below, we everywhere assume that the natural series begins with one.

Assume that --- is a divisible group and --- is the maximum subgroup in . Consider

Prove that --- is a subgroup of containing . Due to the maximum, only two cases are possible: or .

Consider each of the cases separately and come to a contradiction. If so, take it and prove that

is a proper subgroup of , containing and not equal to . In the case, fix and , such as and show that

is a proper subgroup of , containing and not the same as .

Added after 10 minutes 17 seconds:

fsb4000 wrote:

b) give examples of divisible abelian groups, can they be finite?

The simplest example is . Well, or --- whatever you like best.

As for finiteness... of course, a divisible group cannot be finite (except for the trivial case when the group consists of one zero). Assume that --- is a finite group. Prove that for some and all . Then take this and see that the equation is unsolvable for non-zero .

Added after 9 minutes 56 seconds:

fsb4000 wrote:

4. Construct an example of a commutative and associative ring R ()(), in which there are no maximal ideals.

Take an abelian group. Show that it is divisible. Set the multiplication as follows:

Show what for everything that needs to be done is done.

Oops! .. But I made a mistake here, it seems. There is a maximum ideal, it is equal to . Well, yes, I need to think more ... But I won’t think anything now, but I’d better go to work, to the university. You need to leave at least something for an independent decision!

Added after 10 minutes 29 seconds:

fsb4000 wrote:

1. Prove that an arbitrary ring with unit contains a maximal ideal.

according to the solution: 1. By the Zorn lemma, we choose the minimal positive element, and it will be the generating ideal.

Well ... I don’t know what kind of minimal positive element you came up with. In my opinion, this is complete nonsense. What kind of "positive element" will you find in an arbitrary ring, if the order is not specified in this ring and it is not clear what is "positive" and what is "negative" ...

But about the fact that it is necessary to apply the Zorn lemma --- this is the right idea. Only it must be applied to the set of proper ideals of the ring. Take this set, order it by the usual inclusion relation, and show that this ordering is inductive. Then, by Zorn's lemma, you conclude that this set has a maximum element. This maximum element will be the maximum ideal!

When you show inductance, then take their union as the upper bound for the chain of your own ideals. It will also be an ideal, and it will turn out to be its own because the unit will not enter into it. And now, by the way, in a ring without a unit, the proof does not pass through the Zorn lemma, but the whole point is precisely in this moment

Added after 34 minutes 54 seconds:

Alexiii wrote:

Any ring, by definition, has a unit, so it is unthinkable to write "a ring with a unit". Any ring in itself is an ideal of a ring and, moreover, obviously, a maximum ...

We were taught that the presence of a unit is not included in the definition of a ring. So an arbitrary ring does not have to contain a unit, and if it does have one, then it is more than appropriate to say about such a ring that it is a "ring with a unit"!

I think that rummaging in the library, I will find a bunch of very solid algebra textbooks that confirm my point. And in the encyclopedia it is written that the ring is not obliged to have a unit. So everything in the condition of the problem from the author of the topic is correct, there is nothing to drive on him!

The maximal ideal of a ring, by definition, is the ideal that is maximal with respect to inclusion among your own ideals. This is not only written about in many, but simply in all textbooks on algebra, in which the theory of rings is present. So what about the maximum you have one more rut completely off topic!

Added after 6 minutes 5 seconds:

Alexiii wrote:

In general, as I understand from your comments, "rings with unity" are written only in order to exclude a single-element case.

Completely misunderstood! "Rings with a unit" is written in order to indicate the presence of a unit in the ring

And there are plenty of rings without a unit. For example, the set of even integers with the usual addition and multiplication form such a ring.

In various branches of mathematics, as well as in the application of mathematics in technology, there is often a situation where algebraic operations are performed not on numbers, but on objects of a different nature. For example, matrix addition, matrix multiplication, vector addition, operations on polynomials, operations on linear transformations, etc.

Definition 1. A ring is a set of mathematical objects in which two actions are defined - "addition" and "multiplication", which compare ordered pairs of elements with their "sum" and "product", which are elements of the same set. These actions meet the following requirements:

1.a+b=b+a(commutativity of addition).

2.(a+b)+c=a+(b+c)(associativity of addition).

3. There is a zero element 0 such that a+0=a, for any a.

4. For anyone a there is an opposite element − a such that a+(−a)=0.

5. (a+b)c=ac+bc(left distributivity).

5".c(a+b)=ca+cb(right distributivity).

Requirements 2, 3, 4 mean that the set of mathematical objects forms a group , and together with item 1 we are dealing with a commutative (Abelian) group with respect to addition.

As can be seen from the definition, in the general definition of a ring, no restrictions are imposed on multiplications, except for distributivity with addition. However, in various situations, it becomes necessary to consider rings with additional requirements.

6. (ab)c=a(bc)(associativity of multiplication).

7.ab=ba(commutativity of multiplication).

8. Existence of the identity element 1, i.e. such a 1=1 a=a, for any element a.

9. For any element of the element a there is an inverse element a−1 such that aa −1 =a −1 a= 1.

In various rings 6, 7, 8, 9 can be performed both separately and in various combinations.

A ring is called associative if condition 6 is satisfied, commutative if condition 7 is satisfied, commutative and associative if conditions 6 and 7 are satisfied. A ring is called a ring with a unit if condition 8 is satisfied.

Ring examples:

1. Set of square matrices.

Really. The fulfillment of points 1-5, 5 "is obvious. The zero element is the zero matrix. In addition, point 6 (associativity of multiplication), point 8 (the unit element is the identity matrix) are performed. Points 7 and 9 are not performed because in the general case, multiplication of square matrices is non-commutative, and also there is not always an inverse to a square matrix.

2. The set of all complex numbers.

3. The set of all real numbers.

4. The set of all rational numbers.

5. The set of all integers.

Definition 2. Any system of numbers containing the sum, difference and product of any two of its numbers is called number ring.

Examples 2-5 are number rings. Numeric rings are also all even numbers, as well as all integers divisible without remainder by some natural number n. Note that the set of odd numbers is not a ring since the sum of two odd numbers is an even number.

is called the order of the element a. If such n does not exist, then the element a is called an element of infinite order.

Theorem 2.7 (Fermat's little theorem). If a G and G is a finite group, then a |G| =e .

Accept without proof.

Recall that each group G, ° is an algebra with one binary operation for which three conditions are satisfied, i.e., the specified axioms of the group.

A subset G 1 of a set G with the same operation as in a group is called a subgroup if G 1 , ° is a group.

It can be proved that a non-empty subset G 1 of the set G is a subgroup of the group G, ° if and only if the set G 1 together with any elements a and b contains the element a° b -1 .

We can prove the following theorem.

Theorem 2.8. A subgroup of a cyclic group is cyclic.

§ 7. Algebra with two operations. Ring

Consider algebras with two binary operations.

A ring is a non-empty set R, on which two binary operations + and ° are introduced, called addition and multiplication, such that:

1) R; + is an abelian group;

2) multiplication is associative, i.e. for a,b,c R: (a ° b ° ) ° c=a ° (b ° c) ;

3) multiplication is distributive with respect to addition, i.e. for

a,b,c R: a° (b+c)=(a° b)+(a° c) and (a + b)° c= (a° c)+(b° c).

A ring is called commutative if for a,b R: a ° b=b ° a .

The ring is written as R; +, ° .

Since R is an Abelian (commutative) group with respect to addition, it has an additive unit, which is denoted by 0 or θ and is called zero. The additive inverse for a R is denoted by -a. Moreover, in any ring R we have:

0 +x=x+ 0 =x, x+(-x)=(-x)+x=0 , -(-x)=x.

Then we get that

x° y=x° (y+ 0 )=x° y+ x° 0 x° 0 =0 for x R; x° y=(х + 0 )° y=x° y+ 0 ° y 0 ° y=0 for y R.

So, we have shown that for x R: x ° 0 \u003d 0 ° x \u003d 0. However, from the equality x ° y \u003d 0 it does not follow that x \u003d 0 or y \u003d 0. Let's show this with an example.

Example. Let us consider a set of functions that are continuous on an interval. Let us introduce for these functions the usual operations of addition and multiplication: f(x)+ ϕ (x) and f(x) · ϕ (x) . It is easy to see that we get a ring, which is denoted by C . Consider the function f(x) and ϕ (x) shown in Figs. 2.3. Then we get that f(x) ≡ / 0 and ϕ (x) ≡ / 0, but f(x) · ϕ (x) ≡0.

We proved that the product is equal to zero if one of the factors is equal to zero: a ° 0= 0 for a R and showed by example that it can be that a ° b= 0 for a ≠ 0 and b ≠ 0.

If in the ring R we have that a ° b = 0, then a is called left and b right zero divisors. The element 0 is considered a trivial zero divisor.

				f(x) ϕ(x)≡0
	ϕ(x)

A commutative ring without zero divisors other than the trivial zero divisor is called an integral ring or an integral domain.

It is easy to see that

0 =x° (y+(-y))=x° y+x° (-y), 0 =(x+(-x))° y=x° y+(-x)° y

and so x ° (-y)=(-x) ° y is the inverse of the element x° y, i.e.

x ° (-y) \u003d (-x) ° y \u003d - (x ° y).

Similarly, it can be shown that (- x) ° (- y) \u003d x ° y.

§ 8. Ring with unity

If in the ring R there exists a unit with respect to multiplication, then this multiplicative unit is denoted by 1.

It is easy to prove that the multiplicative unit (as well as the additive unit) is unique. The multiplicative inverse for a R (the inverse of multiplication) will be denoted by a-1 .

Theorem 2.9. The elements 0 and 1 are different elements of the nonzero ring R .

Proof. Let R contain not only 0. Then for a ≠ 0 we have a° 0= 0 and a° 1= a ≠ 0, whence it follows that 0 ≠ 1, because if 0= 1, then their products by a would coincide .

Theorem 2.10. Additive unit, i.e. 0 has no multiplicative inverse.

Proof. a° 0= 0° a= 0 ≠ 1 for a R . Thus, a non-zero ring will never be a group with respect to multiplication.

The characteristic of a ring R is the smallest natural number k

such that a + a + ... + a = 0 for all a R . Ring characteristic

k - times

is written k=char R . If the specified number k does not exist, then we set char R= 0.

Let Z be the set of all integers;

Q is the set of all rational numbers;

R is the set of all real numbers; C is the set of all complex numbers.

Each of the sets Z, Q, R, C with the usual operations of addition and multiplication is a ring. These rings are commutative, with a multiplicative unit equal to the number 1. These rings do not have zero divisors, hence they are domains of integrity. The characteristic of each of these rings is equal to zero.

The ring of functions continuous on (ring C ) is also a ring with a multiplicative identity, which coincides with a function that is identically equal to one on . This ring has zero divisors, so it is not an integrity region and char C= 0.

Let's consider one more example. Let M be a non-empty set and R= 2M the set of all subsets of the set M. We introduce two operations on R: the symmetric difference A+ B= A B (which we call addition) and the intersection (which we call multiplication). You can make sure you get

unit ring; the additive unit of this ring will be, and the multiplicative unit of the ring will be the set M. For this ring, for any А, А R , we have: А+ А = А А= . Therefore, charR = 2.

§ 9. Field

A field is a commutative ring whose nonzero elements form a commutative group under multiplication.

We give a direct definition of the field, listing all the axioms.

A field is a set P with two binary operations "+" and "°", called addition and multiplication, such that:

1) addition is associative: for a, b, c R: (a+b)+c=a+(b+c) ;

2) there is an additive unit: 0 P, which for a P: a+0 =0 +a=a;

3) there is an inverse element by addition: for aP(-a)P:

(-a)+a=a+(-a)=0;

4) addition is commutative: for a, b P: a+b=b+a ;

(axioms 1–4 mean that the field is an abelian group by addition);

5) multiplication is associative: for a, b, c P: a ° (b ° c)=(a ° b) ° c ;

6) there is a multiplicative unit: 1 P , which for a P:

1°a=a° 1=a;

7) for any non-null element(a ≠ 0) there is an inverse by multiplication: for a P, a ≠ 0, a -1 P: a -1 ° a = a ° a -1 = 1;

8) multiplication is commutative: for a,b P: a ° b=b ° a ;

(axioms 5–8 mean that a field without a zero element forms a commutative group by multiplication);

9) multiplication is distributive with respect to addition: for a, b, c P: a° (b+c)=(a° b)+(a° c), (b+c) ° a=(b° a)+(c° a).

Field examples:

1) R;+, - field of real numbers;

2) Q;+, - the field of rational numbers;

3) C;+, - the field of complex numbers;

4) let P 2 \u003d (0.1). We define that 1 +2 0=0 +2 1=1,

1 +2 1=0, 0 +2 0=0, 1×0=0×1=0×0=0, 1×1=1. Then F 2 = P 2 ;+ 2 is a field and is called binary arithmetic.

Theorem 2.11. If a ≠ 0, then the equation a ° x \u003d b is uniquely solvable in the field.

Proof . a° x=b a-1° (a° x)=a-1° b (a-1° a)° x=a-1° b

Definition 4.1.1. Ring (K, +, ) is an algebraic system with a non-empty set K and two binary algebraic operations on it, which we will call addition And multiplication. The ring is an Abelian additive group, and multiplication and addition are related by distributive laws: ( a + b)  c = a  c + b  c And from  (a + b) = c  a + c  b for arbitrary a, b, c  K.

Example 4.1.1. We give examples of rings.

1. (Z, +, ), (Q, +, ), (R, +, ), (C, +, ) are the rings of integer, rational, real and complex numbers, respectively, with the usual operations of addition and multiplication. These rings are called numerical.

2. (Z/ nZ, +, ) is the ring of residue classes modulo n  N with operations of addition and multiplication.

3. Lots of M n (K) of all square matrices of fixed order n  N with coefficients from the ring ( K, +, ) with operations of matrix addition and multiplication. In particular, K can be equal Z, Q, R, C or Z/nZ at n  N.

4. The set of all real functions defined on a fixed interval ( a; b) real number axis, with the usual operations of addition and multiplication of functions.

5. Set of polynomials (polynomials) K[x] with coefficients from the ring ( K, +, ) from one variable x with natural operations of addition and multiplication of polynomials. In particular, the rings of polynomials Z[x], Q[x], R[x], C[x], Z/nZ[x] at n  N.

6. Ring of vectors ( V 3 (R), +, ) with addition and vector multiplication.

7. Ring ((0), +, ) with addition and multiplication operations: 0 + 0 = 0, 0  0 = = 0.

Definition 4.1.2. Distinguish finite and endless rings (according to the number of elements of the set K), but the main classification is based on the properties of multiplication. Distinguish associative rings when the multiplication operation is associative (items 1–5, 7 of Example 4.1.1) and non-associative rings (item 6 of example 4.1.1: here , ). Associative rings are divided into unit rings(there is a neutral element with respect to multiplication) and without unit, commutative(the operation of multiplication is commutative) and noncommutative.

Theorem4.1.1. Let be ( K, +, ) is an associative ring with unit. Then the set K* reversible under multiplication of ring elements K is a multiplicative group.

Let us check the fulfillment of the group definition 3.2.1. Let be a, b  K*. Let us show that a  b  K * .  (a  b) –1 = b –1  but –1  K. Really,

(a  b)  (b –1  but –1) = a  (b  b –1)  but –1 = a  1  but –1 = 1,

(b –1  but –1)  (a  b) = b –1  (but –1  a)  b = b –1  1  b = 1,

where but –1 , b –1  K are inverse elements to a And b respectively.

1) Multiplication in K* associative, since K is an associative ring.

2) 1 –1 = 1: 1  1 = 1  1  K* , 1 is a neutral element with respect to multiplication in K * .

3) For  a  K * , but –1  K* , because ( but –1)  a= a  (but –1) = 1
(but –1) –1 = a.

Definition 4.1.3. Lots of K* invertible with respect to multiplication of elements of the ring ( K, +, ) are called multiplicative group of the ring.

Example 4.1.2. Let us give examples of multiplicative groups of various rings.

1. Z * = {1, –1}.

2. M n (Q) * = GL n (Q), M n (R) * = GL n (R), M n (C) * = GL n (C).

3. Z/nZ* is the set of reversible residue classes, Z/nZ * = { | (k, n) = 1, 0  k < n), at n > 1 | Z/nZ * | =  (n), where  is the Euler function.

4. (0) * = (0), since in this case 1 = 0. 

Definition 4.1.4. If in the associative ring ( K, +, ) with unit group K * = K\(0), where 0 is a neutral element with respect to addition, then such a ring is called body or algebra withdivision. The commutative body is called field.

From this definition it is clear that in the body K*   and 1  K* , so 1  0, so the minimal body, which is a field, consists of two elements: 0 and 1.

Example 4.1.3.

1. (Q, +, ), (R, +, ), (C, +, ) are, respectively, the numerical fields of rational, real, and complex numbers.

2. (Z/pZ, +, ) is the final field from p elements, if p- Prime number. For example, ( Z/2Z, +, ) is the minimum field of two elements.

3. A non-commutative body is the body of quaternions - a collection of quaternions, that is, expressions of the form h= a + bi + cj + dk, where a, b, c, d  R, i 2 = = j 2 = k 2 = –1, i  j= k= – j  i, j  k= i= – k  j, i  k= – j= – k  i, with the operations of addition and multiplication. Quaternions are added and multiplied term by term, taking into account the above formulas. For everyone h 0 the inverse quaternion has the form:
.

There are rings with zero divisors and rings without zero divisors.

Definition 4.1.5. If there are non-zero elements in the ring a And b such that a  b= 0, then they are called zero divisors, and the ring itself zero divisor ring. Otherwise, the ring is called ring without zero divisors.

Example 4.1.4.

1. Rings ( Z, +, ), (Q, +, ), (R, +, ), (C, +, ) are rings without zero divisors.

2. in the ring ( V 3 (R), +, ) each non-zero element is a zero divisor, since
for all
V 3 (R).

3. In the ring of matrices M 3 (Z) examples of zero divisors are matrices
And
, because A  B = O(zero matrix).

4. in the ring ( Z/ nZ, +, ) with composite n= k  m, where 1< k, m < n, residue classes And are zero divisors, since . 

Below we present the main properties of rings and fields.

GROUP DEFINITION AND EXAMPLES.

ODA1.Let G be a non-empty set of elements of arbitrary nature. G is called group

1) Bao ° is given on the set G.

2) bao ° is associative.

3) There is a neutral element nÎG.

4) For any element of G, an element symmetric to it always exists and also belongs to G.

Example. The set of Z-numbers with the + operation.

ODA2.Group called abelian, if it is commutative with respect to the given bao °.

Group examples:

1) Z,R,Q "+" (Z+)

The simplest properties of groups

There is only one neutral element in the group

In the group for each element there is a single element symmetrical to it

Let G be a group with bao °, then equations of the form:

a°x=b and x°a=b (1) are solvable and have a unique solution.

Proof. Consider equations (1) for x. Obviously, for a $! a". Since the operation ° is associative, it is obvious that x=b°a" is the only solution.

34. PARITY OF SUBSTITUTION*

Definition 1. The substitution is called even if it decomposes into a product of an even number of transpositions, and odd otherwise.

Suggestion 1.Substitution

Is even<=>- an even permutation. Therefore, the number of even permutations

out of n numbers is equal to n!\2.

Suggestion 2. The permutations f and f - 1 have the same parity character.

> It suffices to check that if is a product of transpositions, then<

Example:

SUBGROUP. SUB-GROUP CRITERION.

Def. Let G be a group with bao ° and a non-empty subset of HÌG. Then H is called a subgroup of G if H is a subgroup with respect to bao° (that is, ° is bao on H. And H with this operation is a group).

Theorem (subgroup criterion). Let G be a group under the operation°, ƹHÎG. H is a subgroup<=>"h 1 ,h 2 нH the condition h 1 °h 2 "нH is satisfied (where h 2 "is a symmetrical element to h 2).

Doc. =>: Let H be a subgroup (we need to prove that h 1 °h 2 "нH). Take h 1 ,h 2 нH, then h 2 "нH and h 1 °h" 2 нH (because h" 2 is a symmetric element to h 2).

<=: (we must prove that H is a subgroup).

Since H¹Æ , then there is at least one element there. Take hнH, n=h°h"нH, i.e., the neutral element nнH. As h 1 we take n, and as h 2 we take h then h"нH Þ "hнH the symmetric element to h also belongs to H.

Let us prove that the composition of any elements from H belongs to H.

Take h 1 , and as h 2 we take h" 2 Þ h 1 °(h 2 ") " нH, Þ h 1 °h 2 нH.

Example. G=S n , n>2, α - some element from Х=(1,…,n). As H we take a non-empty set H= S α n =(fО S n ,f(α)=α), under the action of the mapping from S α n α remains in place. We check the criteria. Take any h 1 ,h 2 ОH. Product h 1 . h 2 "нH, i.e. H is a subgroup, which is called the stationary subgroup of the element α.

RING, FIELD. EXAMPLES.

Def. Let be TO non-empty set with two algebraic operations: addition and multiplication. TO called ring if the following conditions are met:

1) TO - an abelian group (commutative with respect to a given bao °) with respect to addition;

2) multiplication is associative;

3) multiplication is distributive with respect to addition().

If multiplication is commutative, then TO called commutative ring. If there is a neutral element with respect to multiplication, then TO called unit ring.

Examples.

1) The set Z of integers forms a ring with respect to the usual operations of addition and multiplication. This ring is commutative, associative, and has a unit.

2) The sets Q of rational numbers and R of real numbers are fields

about the usual operations of addition and multiplication of numbers.

The simplest properties of rings.

1. Since TO abelian group with respect to addition, then on TO the simplest properties of groups are transferred.

2. Multiplication is distributive with respect to difference: a(b-c)=ab-ac.

Proof. Because ab-ac+ac=ab and a(b-c)+ac=a((b-c)+c)=a(b-c+c)=ab, then a(b-c)=ab-ac.

3. There can be zero divisors in the ring, i.e. ab=0, but it does not follow that a=0 b=0.

For example, in the ring of matrices of size 2´2, there are non-zero elements such that their product will be zero: , where - plays the role of the zero element.

4. a 0=0 a=0.

Proof. Let 0=b-b. Then a(b-b)=ab-ab=0. Similarly, 0 a=0.

5. a(-b)=(-a) b=-ab.

Proof: a(-b)+ab=a((-b)+b)=a 0=0.

6. If in the ring TO there is a unit and it consists of more than one element, then the unit is not equal to zero, where 1 is a neutral element in multiplication; 0 ─ neutral element in addition.

7. Let TO ring with unit, then the set of invertible elements of the ring form a group under multiplication, which is called the multiplicative group of the ring K and denote K*.

Def. A commutative ring with identity, containing at least two elements, in which every non-zero element is invertible, is called field.

The simplest field properties

1. Because the field is a ring, then all the properties of the rings are transferred to the field.

2. There are no zero divisors in the field, i.e. if ab=0 , then a=0 or b=0.

Proof.

If a¹0 , then $ a -1 . Consider a -1 (ab)=(a -1 a)b=0 , and if a¹0 , then b=0, similarly if b¹0

3. An equation of the form a´x=b, a¹0, b - any, in the field has a unique solution x= a -1 b, or x=b/a.

The solution to this equation is called partial.

Examples. 1)PÌC, P - numeric field. 2)P=(0;1);