Mathematical rings. The concept of a ring, the simplest properties of rings
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 there 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.
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 unity, 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);
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.
Non-empty set TO, on which two binary operations are set - addition (+) and multiplication ( ), satisfying the conditions:
1) regarding the operation of addition TO- commutative group;
2) regarding the operation of multiplication TO- semigroup;
3) the operations of addition and multiplication are related by the law of distributivity, i.e. . (a+b)c=ac+bc, c(a+b)=ca+cb for all a, b, c K, is called ring (K,+, ).
Structure (TO,+) is called additive group rings. If the operation of multiplication is commutative, i.e. ab=ba. for all but, b, then the ring is called commutative.
If with respect to the operation of multiplication there is an identity element, which in the ring is usually denoted by the unit 1,. then they say that TO eat unit ring.
A subset L of a ring is called subring, if L is a subgroup of the additive group of the ring, and L is closed under the operation of multiplication, i.e., for all a, b L is running a+b L And ab L.
The intersection of subrings will be a subring. Then, as in the case of groups, by a subring, generated many S K, is called the intersection of all subrings TO, containing S.
1. The set of integers with respect to the operations of multiplication and addition is a (Z, +, )-commutative ring. Sets nZ whole numbers divisible by P, will be a subring without unity for n>1.
Similarly, the set of rational and real numbers are commutative rings with identity.
2. The set of square matrices of order P with respect to the operations of addition and multiplication of matrices, there is a ring with identity E- identity matrix. At n>1 it is non-commutative.
3. Let K-arbitrary commutative ring. Consider all possible polynomials
with variable X and coefficients a 0, a 1, a 2,..., and n, from TO. With respect to the algebraic operations of addition and multiplication of polynomials, this is a commutative ring. It's called polynomial ring K from a variable X over the ring TO(for example, over the ring of integer, rational, real numbers). The ring of polynomials is defined similarly K from T variables as a polynomial ring in one variable x t over the ring K.
4. Let X- arbitrary set, TO- arbitrary ring. Consider the set of all functions f: X K, defined on the set X with values in TO We define the sum and product of functions, as usual, by the equalities
(f+g)(x)=f(x)+g(x); (fg)(x)=f(x)g(x),
where + and - operations in the ring TO.
It is easy to check that all the conditions included in the definition of a ring are satisfied, and the constructed ring will be commutative if the original ring is commutative K. It's called function ring on the set X with values in the ring TO.
Many properties of rings are reformulated corresponding properties of groups and semigroups, for example: a m a n = a m + n, (a t) n = a tp for all m, n and all a.
Other specific properties of rings model properties of numbers:
1) for everyone a a 0=0 a=0;
2) .(-a)b=a(-b)=-(ab);
3) - a=(-1)a.
Really:
2) 0=a(similar to (-a)b=-(ab));
3) using the second property, we have- a= (-a)1 =a(-1) = (-1)a.
Field
In the rings of integers, rational and real numbers from the fact that the product ab=0, it follows that either but=0, or b=0. But in the ring of square matrices of order n>1 this property is no longer satisfied, because, for example, = .
If in the ring K ab=0 at but 0, b, then but is called left b- right zero divisor. If in TO there are no zero divisors (except the element 0, which is a trivial zero divisor), then K called a ring no zero divisors.
1. In the ring function f: R R on the set of real numbers R consider the functions f 1 (x)=|x|+x; f 2 (x) =|x|-x. For them f 1 (x)=0 at x And f2(x)=0 at x, and therefore the product f 1 (x) f 2 (x) is a null function though f 1 (x) And f2(x). Therefore, there are zero divisors in this ring.
2. Consider the set of pairs of integers ( a, b) in which the operations of addition and multiplication are given:
(a 1 , b 1)+(a 2 , b 2)=(a 1 +a 2 , b 1 +b 2);
(a 1 , b 1)(a 2 , b 2)= (a 1 a 2 , b 1 b 2).
This set forms a commutative ring with unity (1,1) and zero divisors, since (1,0)(0,1)=(0,0).
If there are no zero divisors in the ring, then the reduction law is satisfied in it, i.e. ab=ac, a=c. Really, ab-ac=0 a(b-c)=0 (b-c)=0 b=c.
Let be TO- a ring, with a unit. Element but called reversible if there is such an element a -1 , for which aa -1 =a -1 a=1.
The reversible element cannot be a zero divisor, since. if ab=0 , then a -1 (ab) =0 (a -1 a)b=0 1b=0 b=0(similar ba=0 ).
Theorem. All invertible elements of the ring K with identity form a group with respect to multiplication.
Indeed, the multiplication TO associatively, the unit is contained in the set of invertible elements and the product does not infer from the set of invertible elements, since if but And b reversible, then
(ab) -1 = b -1 a -1 .
An important algebraic structure is formed by the commutative rings TO, in which each nonzero element is invertible, i.e., with respect to the operation of multiplication, the set K\(0) forms a group. Three operations are defined in such rings: addition, multiplication, and division.
commutative ring R with unity 1 0, in which every nonzero element is invertible, is called field.
With respect to multiplication, all non-zero elements of the field form a group called multiplicative group fields.
Work ab -1 is written as a fraction and makes sense only when b 0. The element is the only solution to the equation bx=a. Actions with fractions obey the rules familiar to us:
Let us prove, for example, the second of them. Let be x= And y=- solving equations bx=a, dy=c. From these equations it follows dbx=da, bdy=bc bd(x+y)=da+bc t= is the only solution to the equation bdt=da+bc.
1. The ring of integers does not form a field. The field is the set of rational numbers and the set of real numbers.
8.7. Assignments for independent work in chapter 8
8.1. Determine whether the operation of finding the scalar product of vectors in an n-dimensional Euclidean space is commutative and associative. Justify your answer.
8.2. Determine whether the set of square matrices of order n with respect to the operation of matrix multiplication is a group or a monoid.
8.3. Indicate which of the following sets form a group with respect to the operation of multiplication:
a) a set of integers;
b) the set of rational numbers;
c) the set of real numbers other than zero.
8.4. Determine which of the following structures forms a set of square matrices of order n with determinant equal to one: relative to the usual operations of addition and multiplication of matrices:
a) a group
b) ring;
8.5. Indicate what structure the set of integers forms with respect to the operation of multiplication and addition:
a) non-commutative ring;
b) a commutative ring;
8.6. Which of the following structures forms a set of matrices of the form with real a and b with respect to the usual operations of matrix addition and multiplication:
a) a ring
8.7. Which number must be excluded from the set of real numbers so that the remaining numbers form a group with respect to the usual multiplication operation:
8.8. Find out which of the following structures forms a set consisting of two elements a and e, with a binary operation defined as follows:
ee=e, ea=a, ae=a, aa=e.
a) a group
b) an abelian group.
8.9. Are even numbers a ring with respect to the usual operations of addition and multiplication? Justify your answer.
8.10. Is a ring a set of numbers of the form a+b , where a and b are any rational numbers, with respect to addition and multiplication? Justify the answer.
Annotation: In this lecture, the concepts of rings are considered. The main definitions and properties of ring elements are given, associative rings are considered. A number of characteristic problems are considered, the main theorems are proved, and problems for independent consideration are given.
Rings
The set R with two binary operations (addition + and multiplication) is called associative ring with unit, if:
If the operation of multiplication 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 operation of addition , is a commutative group (see example 1.9.4, 2)).
We define the operation of multiplication 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 identity C 1 residue ring modulo n ).
Ring properties (R,+,.)
Lemma 1.10.3 (Newton 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). 
Task 1.10.6. Describe all subrings in the residue ring Z n modulo n .
Remark 1.10.7. In the ring Z 10 the elements divisible by 5 form a ring with 1 which is not a subring in Z 10 (these rings have different identity elements).
Definition 1.10.8. If R is a ring and , , ab=0 , then element a is called a left zero divisor in R , element b is called a right zero divisor in R .
Remark 1.10.9. In commutative rings, of course, there is no difference between left and right zero divisors.
Example 1.10.10. Z , Q , R have no zero divisors.
Example 1.10.11. The ring of continuous functions C has zero divisors. Indeed, if
then , , fg=0 .
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 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 It is clear that a nilpotent element is a zero divisor (if n>1, then 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. An element x of the ring R is called idempotent, if x 2 \u003d 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. We denote by U(R) 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 ).
, , implies 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).
. The smallest such natural number n is called degree of nilpotency of an element .
, ). The converse is not true (there are no nilpotent elements in Z 6, but 2 , 3 , 4 are non-zero zero divisors).
