Isomorphism group ring field definition. Homomorphisms of groups, rings, fields

Definition 1.7. Let be ( A, ) and ( B, ) – groups. Display  : A  B called group homomorphism if it preserves the operation, i.e.  x, y  A  (x  y) =  (x) (y).

Definition 1.8. If a (A, + ,  ) and ( B, , ) – rings, then the mapping  : A  B called ring homomorphism if it preserves both operations, i.e.

 x,yA (x+y) =  (x)  (y), x, y A (x y) =  (x)   (y).

Definition 1.9. Injective homomorphisms are called monomorphisms or investments, surjective homomorphisms – epimorphisms or overlays, and the bijectives isomorphisms.

Definition 1.10. If there is a homomorphism of groups or rings  : BUT B, then the groups or rings BUT, AT called isomorphic.

The meaning of isomorphism is that it establishes such a correspondence between elements of isomorphic objects, which shows that isomorphic objects are indistinguishable from the point of view of preserved algebraic operations.

Examples: 1. Identity isomorphism I: A  A , x  A I (x) = x. (A – group or ring).

2. Unit or null epimorphism: if E = {e} – singleton object (identity group or zero ring), then for any group ( A, ) or a ring, an epimorphism O is defined : A  E,  x  A O (x) = e.

3. Natural embeddings of groups and rings: Z Q  R  C.

Properties of homomorphisms

If a  : (A,  )  (B,  ) – group homomorphism, then

1 0 .  (e A) = e B , those.  converts a single element to a single element.

2 0 .  a  A  (a – 1) =  (a) – 1 , those.  translates the inverse element to a in reverse to  ( a).

thirty . In the case of a ring homomorphism  : (A, + ,  )  (B,  ,  ) we get  (0 BUT) = 0 AT ,  (– a) = –  (a).

4 0 . For a ring homomorphism  : (A, +, )  (B, , ) right:

 x, y  A  (x – y) =  (x) –  (y).

5 0 . Field homomorphism  : (A, + ,  )  (B,  ,  ) either null or nesting.

60. If  : u  V and  : V  w are two group or ring homomorphisms, then their composition  ○  : u  w is a group or ring homomorphism.

70. If  : V  w is an isomorphism of groups or rings, then the inverse mapping  –1: w  V is also an isomorphism of groups or rings. The concept and idea of isomorphism in modern mathematics

Isomorphism (or isomorphism) is one of the fundamental concepts of modern mathematics. Two mathematical objects (or structures) of the same type are called isomorphic if there is a one-to-one mapping of one of them onto the other, such that it and its inverse preserve the structure of the objects, i.e. elements that are in some relation are translated into elements that are in the corresponding relation.

Isomorphic objects may have a different nature of elements and relations between them, but they have exactly the same abstract structure, they serve as copies of each other. Isomorphism is an "abstract equality" of objects of the same type. For example, the additive group of residue classes modulo n is isomorphic to the multiplicative group of complex roots n th degree out of 1.

The isomorphism relation on any class of mathematical objects of the same type, being an equivalence relation, splits the original class of objects into isomorphism classes – classes of pairwise isomorphic objects. Choosing one object in each isomorphism class, we obtain a complete abstract overview of this class of mathematical objects. The idea of isomorphism is to represent or describe objects of a given class up to isomorphism.

For each given class of objects, there is isomorphism problem. Are two arbitrary objects from a given class isomorphic? How is it found out? To prove the isomorphism of two objects, as a rule, a specific isomorphism is constructed between them. Or it is established that both objects are isomorphic to some third object. To check that two objects are not isomorphic, it suffices to specify an abstract property that one of the objects has but the other does not.

METHOD 11. Yu.M. Kolyagin distinguishes between two types of extracurricular work in mathematics.

Work with students lagging behind others in the study of program material, i.e. additional lessons in mathematics.

Working with students who are interested in mathematics.

But there is also a third type of work.

Working with students to develop interest in learning mathematics.

There are the following forms of extracurricular work:

Mathematical circle.

Optional.

Olympiad contests, quizzes.

Mathematical Olympiads.

Mathematical discussions.

Math Week.

School and classroom math print.

Making mathematical models.

Mathematical excursions.

These forms often intersect and therefore it is difficult to draw sharp boundaries between them. Moreover, elements of many forms can be used in organizing work on any one of them. For example, when holding a mathematical evening, you can use competitions, contests, reports, etc.

stages of organization.

Preparatory

Organizational

arouse interest in extracurricular activities;

attract to participate in public events and individual competitions;

Didactic

help in overcoming difficulties;

support the emerging interest in additional activities;

desire to engage in mathematical self-education

Basic

create a base for each student for further personal success;

to help students realize the social, practical and personal significance of extracurricular activities;

to form a positive motivation for participation in extracurricular activities

Final

to carry out diagnostics and reflection of extra-curricular activities;

sum up and reward students who took an active part

The fact that the concept of isomorphism really expresses the identity of all the considered properties of sets can be formulated as the following proposition:

If the sets M and M" are isomorphic with respect to some system of relations S, then any property of the set M, formulated in terms of the relations of the system S(and, hence, the relations defined through the relations of the system S) is carried over to the set M", and back.

Let's analyze this situation with a specific example.

Let in sets M and M" the relation "greater than" is defined, and they are isomorphic with respect to this relation; then if M ordered, i.e. if in M properties 1) and 2) from section are satisfied, then they are also satisfied in M".

Let us prove property 1). Let be a" and b"- elements M" and a and b- relevant elements M. By virtue of condition 1) in M one of the relations a = b, a > b, b > a. Display M on the M" retains the "greater than" relationship. So, one of the relations a" = b", a" > b", b" > a". If in M" performed more than one of them, then from saving the "greater than" relation when displaying M" on the M there should be more than one relation for a and b, which contradicts condition 1).

Let us prove property 2). If a a" > b" and b" > c", then also a > b and b > c. Indeed, in M should be a > c. Means, a" > c".

Let us now deal with the isomorphism of groups of rings and fields. Since the relationship here a + b = c and ab = c satisfy the additional requirement that for any a and b there is one and only one c, for which a + b = c or ab = c(these two requirements are essentially two additional axioms), and these requirements are assumed to be satisfied as in M, as well as in M", the definition of an isomorphism of groups of rings and fields can be simplified in comparison with the definition of , namely, to require that the basic relations be preserved only when passing from M to M". Restricting ourselves to the case of rings and fields, which will be needed later in the determination of numerical domains (the case of groups differs from the one considered only in that there is one operation instead of two), we thus obtain:

Ring (or field) R called ring isomorphic(respectively field) R"(record ) if there is a one-to-one mapping R on the R", at which the sum and product of any elements R match the sum and product of the corresponding elements R".

Let us show that this definition is a special case of the general definition . To do this, we only need to make sure that the inverse mapping R" on the R also saves sum and product. Let in R" we have: a" + b" = c", and elements a", b", c" when reversed, correspond a, b, c from R. We have to prove that a + b = c. But if a + b = d ≠ c, then from the definition given in the previous paragraph it would follow a" + b" = d" ≠ c", which contradicts the uniqueness of the addition operation in R"

Definition 37. Non-empty subset H fields R containing at least two elements is called underground fields R, if H is a field with respect to the same operations as a field R.

Theorem 10(subfield criterion).

Let be R - field, H≠ Æ, ∣ H∣≥2 , H Í R. N is an field subfield R if and only if the following conditions are met:

1) for any h1, h2∈H: h 1 - h 2∈H;

2) for any h 1 , h 2∈H: h 1 h 2∈H;

3) for any h∈H# ∃ h-1∈H#.

Proof. Need. Let be H– field subfield R. Then, by definition 37, H- field. Hence, H is an additive Abelian group. Means, H is closed under the operation of addition and for any h∈H ∃-h∈H, that is, condition 1) is satisfied. Besides, H# is a multiplicative abelian group. Hence, conditions 2) and 3) are satisfied.

Adequacy. Let conditions 1), 2) and 3) be satisfied. Let us show that H– field subfield R. It is enough to show that H- field. Condition 1) implies that H is a subgroup of an additive abelian group R. Hence H is an additive Abelian group. From conditions 2) and 3) we have H# is a subgroup of a multiplicative abelian group P# . So H# is a multiplicative abelian group. In addition, since HÍ R and in R distributive laws hold, then H distributive laws also hold. Thus, H field, and therefore H– field subfield R.

The theorem has been proven.

Definition 38. One-to-one mapping φ fields R on the field R called isomorphic mapping or isomorphism if 2 conditions are met:

1) for any a, b∈Р φ(a+b)=φ (a)+φ (b);

2) for any a, b∈Р φ(a⋅b)=φ (a)⋅φ (b).

End of work -

This topic belongs to:

Elements of set theory The concept of set. Subset. Operations on sets

In the school course of mathematics, operations on numbers were considered. At the same time, a number of properties of these operations were established .. Along with operations on numbers, the school course also considered and .. The main goal of the algebra course is to study algebras and algebraic systems.

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Definition 34. Non-empty subset H rings K called subring rings K, if H is a ring with respect to the same operations as a ring K.

Theorem 9(subring criterion).

Let be K- ring, H- non-empty subset K. H is a subring of the ring K if and only if the following conditions are met:

1) for any h1, h2∈H (h1-h2)∈H;

2) for any h1, h2∈H h 1 ⋅ h 2∈H.

Proof. Need. Let be H- subring of the ring K. Then H is a ring with respect to the same operations as K. Means, H is closed under the operations of addition and multiplication, that is, condition 2) is satisfied. In addition, for any h1, h2∈H∃ -h 2∈H and h1+(-h 2)=h1-h2∈H.

Adequacy. Let conditions 1) and 2) be satisfied. Let's prove that H - subring of the ring K. By Definition 34, it suffices to check that H - ring.

Since condition 1) is satisfied, then, by Theorem 7", H is a subgroup of the additive group K. Moreover, since the operation of addition is commutative on K, then in H the operation "+" is also commutative. Hence, H is an additive Abelian group.

Next, in K distributive laws are obeyed and H⊆K. So in H distributive laws also hold. Thus, we have shown that H is a ring, and therefore H- ring subring K.

The theorem has been proven.

Definition 35. Display φ rings K in the ring K called homomorphic mapping or homomorphism if 2 conditions are met:

1) for any a, b∈K φ(a+b)=φ (a)+φ (b);

2) for any a, b∈K φ(a⋅b)=φ (a)⋅φ (b).

Remark 10. The definitions of monomorphism, epimorphism, isomorphism, endomorphism, automorphism of rings are formulated similarly to the corresponding definitions for groups.

Remark 11. The isomorphism relation on the set of all rings is an equivalence relation that splits the given set into disjoint classes - equivalence classes. One class will include those and only those rings that are isomorphic with each other. Isomorphic rings have the same properties. Therefore, from an algebraic point of view, they are indistinguishable.

8. Field.