Definition and basic properties of rings. Rings. Fields. Ideals and homomorphisms of rings. §5. Field of complex numbers. Operations on complex

Definition 2.5. Ring called algebra

R = (R, +, ⋅, 0 , 1 ),

whose signature consists of two binary and two nullary operations, and for any a, b, c ∈ R the following equalities hold:

1. a+(b+c) = (a+b)+c;
2. a+b = b+a;
3. a + 0 = a;
4. for every a ∈ R there is an element a" such that a+a" = 0
5. a-(b-c) = (a-b)-c;
6. a ⋅ 1 = 1 ⋅ a = a;
7. a⋅(b + c) = a⋅b + a⋅c, (b + c) ⋅ a = b⋅ a + c⋅a.

The operation + is called adding the ring , operation ring multiplication , element 0 - zero of the ring , element 1 - ring unit .

Equalities 1-7 specified in the definition are called axioms of the ring . Let us consider these equalities from the point of view of the concept groups And monoid.

Ring axioms 1-4 mean that the algebra (R, +, 0 ), the signature of which consists only of the operations of addition of the ring + and zero of the ring 0 , is abelian group. This group is called additive group of the ring R and they also say that by addition the ring is a commutative (Abelian) group.

Ring axioms 5 and 6 show that the algebra (R, ⋅, 1), whose signature includes only the multiplication of the ring ⋅ and the identity of the ring 1, is a monoid. This monoid is called multiplicative monoid of the ring R and they say that by multiplication a ring is a monoid.

The connection between ring addition and ring multiplication is established by Axiom 7, according to which the multiplication operation is distributive with respect to the addition operation.

Considering the above, we note that a ring is an algebra with two binary and two nullary operations R =(R, +, ⋅, 0 , 1 ), such that:

1. algebra (R, +, 0 ) - commutative group;
2. algebra (R, ⋅, 1 ) - monoid;
3. the operation ⋅ (multiplication of a ring) is distributive with respect to the operation + (addition of a ring).

Remark 2.2. In the literature there is a different composition of ring axioms related to multiplication. Thus, axiom 6 may be absent (there is no 1 ) and axiom 5 (multiplication is not associative). In this case, associative rings are distinguished (the requirement of associative multiplication is added to the axioms of the ring) and rings with unity. In the latter case, the requirements of associativity of multiplication and the existence of a unit are added.

Definition 2.6. The ring is called commutative , if its multiplication operation is commutative.

Example 2.12. A. The algebra (ℤ, +, ⋅, 0, 1) is a commutative ring. Note that the algebra (ℕ 0, +, ⋅, 0, 1) will not be a ring, since (ℕ 0, +) is a commutative monoid, but not a group.

b. Consider the algebra ℤ k = ((0,1,..., k - 1), ⊕ k , ⨀ k , 0,1) (k>1) with the operation ⊕ k of addition modulo l and ⨀ k (multiplication modulo l). The latter is similar to the operation of addition modulo l: m ⨀ k n is equal to the remainder of division by k of the number m ⋅ n. This algebra is a commutative ring, which is called ring of residues modulo k.

V. The algebra (2 A, Δ, ∩, ∅, A) is a commutative ring, which follows from the properties of intersection and symmetric difference of sets.

G. An example of a non-commutative ring gives the set of all square matrices of a fixed order with the operations of matrix addition and multiplication. The unit of this ring is the identity matrix, and the zero is the zero matrix.

d. Let L- linear space. Let us consider the set of all linear operators acting in this space.

Let us remind you that amount two linear operators A And IN called operator A + B, such that ( A + IN) X = Oh +In, X∈ L.

Product of linear operators A And IN is called a linear-linear operator AB, such that ( AB)X = A(In) for anyone X ∈ L.

Using the properties of the indicated operations on linear operators, we can show that the set of all linear operators acting in space L, together with the operations of addition and multiplication of operators, forms a ring. The zero of this ring is null operator, and by unit - identity operator.

This ring is called ring of linear operators in linear space L. #

The ring axioms are also called basic identities of the ring . A ring identity is an equality whose validity is preserved when any elements of the ring are substituted for the variables appearing in it. Basic identities are postulated, and from them other identities can then be deduced as consequences. Let's look at some of them.

Recall that the additive group of a ring is commutative and the operation is defined in it subtraction.

Theorem 2.8. In any ring the following identities hold:

1. 0 ⋅ a = a ⋅ 0 = 0 ;
2. (-a) ⋅ b = -(a ⋅ b) = a ⋅ (-b);
3. (a-b) ⋅ c = a ⋅ c - b ⋅ c, c ⋅ (a-b) = c ⋅ a - c ⋅ b.

◀Let's prove the identity 0 ⋅ a = 0 . Let us write for arbitrary a:

a+ 0 ⋅ a = 1 ⋅ a + 0 ⋅ a = ( 1 +0 ) ⋅ a = 1 ⋅ a = a

So, a + 0 ⋅ a = a. The last equality can be considered as an equation in the additive group of a ring with respect to an unknown element 0 ⋅ a. Since in the additive group any equation of the form a + x = b has a unique solution x = b - a, then 0 ⋅ a = a - a = 0 . Identity a⋅ 0 = 0 is proved in a similar way.

Let us now prove the identity - (a ⋅ b) = a ⋅ (-b). We have

a ⋅ (-b)+a ⋅ b = a ⋅ ((-b) + b) = a ⋅ 0 = 0 ,

whence a ⋅ (-b) = -(a ⋅ b). In the same way, one can prove that (-a) ⋅ b = -(a ⋅ b).

Let us prove the third pair of identities. Let's consider the first of them. Taking into account what was proved above, we have

a ⋅ (b - c) = a ⋅ (b+(-c)) = a ⋅ b + a ⋅ (-c) =a ⋅ b - a ⋅ c,

those. the identity is true. The second identity of this pair is proved in a similar way.

Corollary 2.1. In any ring the identity ( -1 ) ⋅ x = x ⋅ ( -1 ) = -x.

◀The indicated corollary follows from the second identity of Theorem 2.8 for a = 1 and b = x.

The first two identities proved in Theorem 2.8 express a property called nullifying property of zero in the ring. The third pair of identities of this theorem expresses the distributive property of the operation of multiplication of a ring with respect to the operation of subtraction. Thus, when performing calculations in any ring, you can open the brackets and change the signs in the same way as when adding, subtracting and multiplying real numbers.

Nonzero elements a and b of the ring R called dividers zero , if a ⋅ b = 0 or b ⋅ a = 0 . An example of a ring with a zero divisor gives any modulo residue ring k if k is a composite number. In this case, the product modulo k of any type that yields a multiple of k during ordinary multiplication will be equal to zero. For example, in a residue ring modulo 6, elements 2 and 3 are zero divisors, since 2 ⨀ 6 3 = 0. Another example is given by a ring of square matrices of a fixed order (at least two). For example, for second-order matrices we have

When a and b are non-zero, the given matrices are zero divisors.

By multiplication, a ring is only a monoid. Let us pose the question: in what cases will a multiplication ring be a group? First of all, note that the set of all elements of the ring in which 0 ≠ 1 , cannot form multiplication groups, since zero cannot have an inverse. Indeed, if we assume that such an element 0" exists, then, on the one hand, 0 ⋅ 0" = 0" ⋅ 0 = 1 , and on the other - 0 ⋅ 0" = 0" ⋅ 0 = 0 , from which 0 = 1. This contradicts the condition 0 ≠ 1 . Thus, the question posed above can be refined as follows: in what cases does the set of all non-zero elements of a ring form a group under multiplication?

If a ring has zero divisors, then the subset of all non-zero elements of the ring does not form a multiplication group, if only because this subset is not closed under the multiplication operation, i.e. There are non-zero elements whose product is equal to zero.

A ring in which the set of all non-zero elements by multiplication forms a group is called body , commutative body - field , and the group of non-zero elements of the body (field) by multiplication - multiplicative group this body (fields). According to the definition, a field is a special case of a ring in which operations have additional properties. Let's write down all the properties that are required for field operations. They are also called field axioms .

The field is an algebra F = (F, +, ⋅, 0, 1), the signature of which consists of two binary and two nullary operations, and the identities are valid:

1. a+(b+c) = (a+b)+c;
2. a+b = b+a;
3. a+0 = a;
4. for every a ∈ F there is an element -a such that a+ (-a) = 0;
5. a ⋅ (b ⋅ c) = (a ⋅ b) ⋅ c;
6. a ⋅ b = b ⋅ a
7. a ⋅ 1 = 1 ⋅ a = a
8. for every a ∈ F different from 0, there is an element a -1 such that a ⋅ a -1 = 1;
9. a ⋅ (b+c) = a ⋅ b + a ⋅ c.

Example 2.13. A. Algebra (ℚ, +, ⋅, 0, 1) is a field called field of rational numbers .

b. The algebras (ℝ, +, ⋅, 0, 1) and (ℂ, +, ⋅, 0, 1) are fields called fields of real and complex numbers respectively.

V. An example of a body that is not a field is algebra quaternions . #

So, we see that the field axioms correspond to the known laws of addition and multiplication of numbers. When doing numerical calculations, we “work in the fields,” namely, we deal primarily with the fields of rational and real numbers, sometimes we “move” to the field of complex numbers.

Containing a unit is called ring with one . Unit is usually designated by the number “1” (which reflects the properties of the number of the same name) or sometimes (for example, in matrix algebra), by the Latin letter I or E.

Different definitions of algebraic objects may either require the presence of a unit or leave it as an optional element. A one-sided neutral element is not called a unit. The unit is unique in the general property of a two-sided neutral element.

Sometimes the units of a ring are called its invertible elements, which can cause confusion.

One, zero and category theory

The unit is the only element of the ring that is both idempotent and invertible.

Reversibility

Reversible Any element u of a ring with unity that is a two-sided divisor of unity is called, that is:

while maintaining such properties as associativity and commutativity of multiplication. Element 1 will be the unit of extended algebra. If there was already a unit in the algebra, then after expansion it will turn into an irreversible idempotent.

This can also be done with a ring, for example, because every ring is an associative algebra over

is called the order of 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 .

We will accept without proof.

Recall that each group G, ° is an algebra with one binary operation for which three conditions are satisfied, i.e. the indicated 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 proven that a non-empty subset G 1 of a set G is a subgroup of a group G, ° if and only if the set G 1, together with any elements a and b, contains the element a ° b -1.

The following theorem can be proven.

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

§ 7. Algebra with two operations. Ring

Let's consider algebras with two binary operations.

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

We write the ring as R; +, ° .

Since R is an Abelian (commutative) group under addition, it has an additive unit, which is denoted by 0 or θ and called zero. The additive inverse of 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=(x + 0 )° y=x° y+ 0 ° y 0 ° y=0 for y R.

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

Example. Let us consider a set of functions continuous on an interval. Let us introduce the usual operations of addition and multiplication for these functions: f(x)+ ϕ (x) and f(x)· ϕ (x) . As is easy to see, we get a ring, which is denoted by C. Consider the function f(x) and ϕ (x) shown in Fig. 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 the left and b the right divisors of zero. We consider the element 0 to be a trivial divisor of zero.

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

It's easy to see that

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

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

x ° (-y) = (-x)° y = -(x ° y).

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

§ 8. Ring with unity

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

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

Theorem 2.9. Elements 0 and 1 are distinct 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, which implies that 0 ≠ 1, because if 0= 1, then their products on a would coincide .

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

Proof. а° 0= 0° а= 0 ≠ 1 for а R . Thus, a nonzero ring will never be a group under multiplication.

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

such that a + a + ... + a = 0 for all a R . Characteristics of the ring

k − times

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 – the set of all rational numbers;

R – 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 have no zero divisors, hence they are domains of integrity. The characteristic of each of these rings is zero.

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

Let's look at another example. Let M be a non-empty set and R = 2M the set of all subsets of the set M. Let us introduce two operations on R: the symmetric difference A + B = A B (which we will call addition) and intersection (which we will call multiplication). You can make sure you received

ring with unit; 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 A, A R, we have: A+ A = A A=. Therefore, charR = 2.

§ 9. Field

A field is a commutative ring whose non-zero elements form a commutative group under multiplication.

Let us 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) additive unit exists: 0 P, that for a P: a+0 =0 +a=a;

3) there is an inverse element for addition: for a P (-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 under 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-zero element(a ≠ 0) there is an inverse element of 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 under multiplication);

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

Example fields:

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

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

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

4) let P 2 = (0,1). Let us determine 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=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

Annotation: This lecture discusses the concepts of rings. The basic definitions and properties of ring elements are given, and associative rings are considered. A number of characteristic problems are considered, the main theorems are proved, and problems for independent consideration are given

Rings

A set R with two binary operations (addition + and multiplication) is called associative ring with unit, If:

If the multiplication operation is commutative, then the ring is called commutative ring. Commutative rings are one of the main objects of study in commutative algebra and algebraic geometry.

Notes 1.10.1.

Examples 1.10.2 (examples of associative rings).

We have already seen that the group of residues (Z n ,+)=(C 0 ,C 1 ,...,C n-1 ), C k =k+nZ, modulo n with the addition operation, is a commutative group (see example 1.9.4, 2)).

Let us define the multiplication operation by setting . Let's check the correctness of this operation. If C k =C k" , C l =C l" , then k"=k+nu , l"=l+nv , and therefore C k"l" =C kl .

Because (C k C l)C m =C (kl)m =C k(lm) =C k (C l C m), C k C l =C kl =C lk =C l C k , C 1 C k =C k =C k C 1 , (C k +C l)C m =C (k+l)m =C km+lm =C k C m +C l C m, then is an associative commutative ring with unit C 1 residue ring modulo n).

Properties of rings (R,+,.)

Lemma 1.10.3 (Newton's binomial). Let R be a ring with 1 , , . Then:

Proof.

Definition 1.10.4. A subset S of a ring R is called subring, If:

a) S is a subgroup with respect to addition in the group (R,+);

b) for we have ;

c) for a ring R with 1 it is assumed that .

Examples 1.10.5 (examples of subrings).

Problem 1.10.6. Describe all subrings in the residue ring Zn modulo n.

Note 1.10.7. In the ring Z 10, elements that are multiples of 5 form a ring with 1, which is not a subring in Z 10 (these rings have different unit elements).

Definition 1.10.8. If R is a ring, and , ab=0, then the element a is called the left zero divisor in R, the element b is called the right zero divisor in R.

Note 1.10.9. In commutative rings, of course, there is no difference between left and right zero divisors.

Example 1.10.10. There are no zero divisors in Z, Q, R.

Example 1.10.11. The ring of continuous functions C has zero divisors. Indeed, if

Example 1.10.12. If n=kl , 1

Lemma 1.10.13. If there are no (left) zero divisors in the ring R, then from ab=ac , where , , it follows that b=c (i.e., the ability to cancel by a non-zero element on the left if there are no left zero divisors; and on the right if there are no right zero divisors).

Proof. If ab=ac , then a(b-c)=0 . Since a is not a left zero divisor, then b-c=0, i.e. b=c.

Definition 1.10.14. The element is called nilpotent, if x n =0 for some . The smallest natural number n is called degree of nilpotency of an element .

It is clear that a nilpotent element is a zero divisor (if n>1 then , ). The converse statement is not true (there are no nilpotent elements in Z 6, but 2, 3, 4 are non-zero divisors of zero).

Exercise 1.10.15. The ring Z n contains nilpotent elements if and only if n is divisible by m 2 , where , .

Definition 1.10.16. The element x of the ring R is called idempotent, if x 2 =x . It is clear that 0 2 =0, 1 2 =1. If x 2 =x and , then x(x-1)=x 2 -x=0, and therefore non-trivial idempotents are zero divisors.

By U(R) we denote the set of invertible elements of the associative ring R, i.e. those for which there is an inverse element s=r -1 (i.e. rr -1 =1=r -1 r ).

Let (K,+, ·) be a ring. Since (K, +) is an Abelian group, taking into account the properties of groups we obtain

SV-VO 1. In every ring (K,+, ·) there is a unique zero element 0 and for every a ∈ K there is a unique element opposite to it -a.

NE-VO 2. ∀ a, b, c ∈ K (a + b = a + c ⇒ b = c).

SV-VO 3. For any a, b ∈ K in the ring K there is a unique difference a − b, and a − b = a + (−b). Thus, the subtraction operation is defined in the ring K, and it has properties 1′-8′.

SV-VO 4. The multiplication operation in K is distributive with respect to the subtraction operation, i.e. ∀ a, b, c ∈ K ((a − b)c = ac − bc ∧ c(a − b) = ca − cb).

Doc. Let a, b, c ∈ K. Taking into account the distributivity of the operation · in K with respect to the operation + and the definition of the difference of elements of the ring, we obtain (a − b)c + bc = ((a − b) + b)c = ac, whence by definition difference it follows that (a − b)c = ac − bc.

The right law of distributivity of the multiplication operation relative to the subtraction operation is proved in a similar way.

SV-V 5. ∀ a ∈ K a0 = 0a = 0.

Proof. Let a ∈ K and a b-arbitrary element from K. Then b − b = 0 and therefore, taking into account the previous property, we obtain a0 = a(b − b) = ab − ab = 0.

It is proved in a similar way that 0a = 0.

NE-VO 6. ∀ a, b ∈ K (−a)b = a(−b) = −(ab).

Proof. Let a, b ∈ K. Then (−a)b + ab = ((−a) + a)b =

0b = 0. Hence, (−a)b = −(ab).

The equality a(−b) = −(ab) is proved in a similar way.

NE-VO 7. ∀ a, b ∈ K (−a)(−b) = ab.

Proof. Indeed, applying the previous property twice, we obtain (−a)(−b) = −(a(−b)) = −(−(ab)) = ab.

COMMENT. Properties 6 and 7 are called the rules of signs in the ring.

From the distributivity of the multiplication operation in the ring K relative to the addition operation and properties 6 and 7, the following follows:

SV-VO 8. Let k, l be arbitrary integers. Then ∀ a, b ∈ K (ka)(lb) = (kl)ab.

Subring

A subring of a ring (K,+, ·) is a subset H of a set K that is closed under the operations + and · defined in K and is itself a ring under these operations.

Examples of subrings:

Thus, Z is a subring of the ring (Q,+, ·), Q is a subring of the ring (R,+, ·), Rn×n is a subring of the ring (Cn×n,+, ·), Z[x] is a subring of the ring ( R[x],+, ·), D is a subring of the ring (C,+, ·).

In any ring (K,+, ·), the set K itself, as well as the singleton subset (0) are subrings of the ring (K,+, ·). These are the so-called trivial subrings of the ring (K,+, ·).

The simplest properties of subrings.

Let H be a subring of the ring (K,+, ·), i.e. (H,+, ·) is itself a ring. This means that the (H, +)-group, i.e. H is a subgroup of the group (K, +). Therefore, the following statements are true.

SV-VO 1. The zero element of the subring H of the ring K coincides with the zero element of the ring K.

SV-VO 2. For any element a of the subring H of the ring K, its opposite element in H coincides with −a, i.e. with its opposite element in K.

SV-VO 3. For any elements a and b of the subring H, their difference in H coincides with the element a − b, i.e. with the difference of these elements in K.

Signs of a subring.

THEOREM 1 (first sign of a subring).

A non-empty subset H of a ring K with the operations + and · is a subring of the ring K if and only if it satisfies the following conditions:

∀ a, b ∈ H a + b ∈ H, (1)

∀ a ∈ H − a ∈ H, (2)

∀ a, b ∈ H ab ∈ H. (3)

Necessity. Let H be a subring of the ring (K,+, ·). Then H is a subgroup of the group (K, +). Therefore, by the first criterion of a subgroup (in the additive formulation), H satisfies conditions (1) and (2). Moreover, H is closed under the multiplication operation defined in K, i.e. H

also satisfies condition (3).

Adequacy. Let H ⊂ K, H 6= ∅ and H satisfies conditions (1) − (3). From conditions (1) and (2) according to the first criterion of a subgroup it follows that H is a subgroup of the group (K, +), i.e. (H, +)-group. Moreover, since (K, +) is an Abelian group, (H, +) is also Abelian. In addition, from condition (3) it follows that multiplication is a binary operation on the set H. The associativity of the operation · in H and its distributivity with respect to the operation + follow from the fact that the operations + and · in K have such properties.

THEOREM 2 (second sign of a subring).

A non-empty subset H of a ring K with the operations + and · is

subring of the ring K t. and t. t, when it satisfies the following conditions:

∀ a, b ∈ H a − b ∈ H, (4)

∀ a, b ∈ H ab ∈ H. (5)

The proof of this theorem is similar to the proof of Theorem 1.

In this case, Theorem 2′ (the second criterion of a subgroup in the additive formulation) and a remark to it are used.

7.Field (definition, types, properties, characteristics).

A field is a commutative ring with identity e is not equal to 0 , in which every element different from zero has an inverse.

Classic examples of number fields are the fields (Q,+, ·), (R,+, ·), (C,+, ·).

PROPERTY 1 . In every field F the law of contraction is valid

by a common factor different from zero, i.e.

∀ a, b, c ∈ F (ab = ac ∧ a is not equal to 0 ⇒ b = c).

PROPERTY 2 . In every field F no zero divisors.

PROPERTY 3 . Ring(K,+, ·) is a field if and only

when there are many K\(0) is a commutative group with respect to the operation of multiplication.

PROPERTY 4 . Finite nonzero commutative ring(K,+, ·) without zero divisors is a field.

The quotient of the field elements.

Let (F,+, ·) be a field.

Partial elements a And b fields F , Where b is not equal to 0 ,

such an element is called c ∈ F , What a = bc .

PROPERTY 1 . For any elements a And b fields F , Where b is not equal to 0 , there is a unique quotient a/b , and a/b= ab−1.

PROPERTY 2 . ∀ a ∈ F \ (0)

a/a=e And∀ a ∈ F a/e= a.

PROPERTY 3 . ∀ a, c ∈ F ∀ b, d ∈ F \ (0)

a/b=c/d ⇔ ad = bc.

PROPERTY 4 . ∀ a, c ∈ F ∀ b, d ∈ F \ (0)

PROPERTY 5 . ∀ a ∈ F ∀ b, c, d ∈ F \ (0)

(a/b)/(c/d)=ad/bc

PROPERTY 6 . ∀ a ∈ F ∀ b, c ∈ F \ (0)

PROPERTY 7 . ∀ a ∈ F ∀ b, c ∈ F \ (0)

PROPERTY 8 . ∀ a, b ∈ F ∀ c ∈ F \ (0)

Field F , whose unit has finite order p in Group(F, +) p .

Field F unit, which has infinite order in the group(F, +) , is called the characteristic field 0.

8. Subfield (definition, types, properties, characteristics)

Field subfield(F,+, ·) called a subset S sets F , which is closed under the operations+ And· , defined in F , and itself is a field relative to these operations.

Let us give some examples of subfields Q-subfield of the field (R,+, ·);

R-subfield of the field (C,+, ·);

The following statements are true.

PROPERTY 1 . Subfield element zero S fields F coincides with

zero element of the field F .

PROPERTY 2 . For every element a subfields S fields F its opposite element in S coincides with−a , i.e. with its opposite element in F .

PROPERTY 3 . For any elements a And b subfields S fields F their

difference in S coincides with a−b those. with the difference of these elements in F .

PROPERTY 4 . Subfield unit S fields F coincides with one

e fields F .

PROPERTY 5 . For every element a subfields S fields F , from-

personal from zero, its inverse element in S coincides with a−1 , i.e. with the element inverse to a V F .

Signs of the subfield.

THEOREM 1 (the first sign of a subfield).

Subset H fields F with operations+, · , containing non-zero

(F,+, ·)

∀ a, b ∈ H a + b ∈ H, (1)

∀ a ∈ H − a ∈ H, (2)

∀ a, b ∈ H ab ∈ H, (3)

∀ a ∈ H \ (0) a−1 ∈ H. (4)

THEOREM2 (second sign of the subfield).

Subset H fields F with operations+, · , containing non-zero

element is a subfield of the field(F,+, ·) if and only if it satisfies the following conditions:

∀ a, b ∈ H a − b ∈ H, (5)

∀ a ∈ H ∀ b ∈ H\(0) a/b ∈ H. (6)

10. Divisibility relation in the Z ring

Statement: for any elements a,b,c of a commutative ring on the set R, the following implications hold:

1) a|b, b|c => a|c

2) a|b, a|c => a| (b c)

3) a|b => a|bc

for any a, b Z the following is true:

2) a|b, b≠0 => |a|≤|b|

3)a|b and b|a ó |a|=|b|

Dividing the integer a by the integer b with the remainder means finding integers q and r such that you can represent a=b*q + r, 0≤r≥|b|, where q is the incomplete quotient, r is the remainder

Theorem: If a and b Z, b≠0, then a can be divided by b with a remainder, and the incomplete quotient and remainder are uniquely determined.

Corollary, if a and b Z , b≠0, then b|a ó

11. GCD and NOC

The greatest common divisor (GCD) of the numbers Z is some number d that satisfies the following conditions

1) d is a common divisor i.e. d| ,d| …d|

2) d is divisible by any common divisor of numbers, i.e. d| ,d| …d| =>d| ,d| …d|