Number ring

Mathematical terminology

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Number ring is a special number set. The ring composed of numbers is ring The most basic examples and models of. Let P be a nonempty subset of the complex number set. If the sum, difference and product of any two numbers in P still belong to P, then P is said to be a Number ring 。 For example, the set Z of all integers, the set Q of all rational numbers, the set R of all real numbers and the set C of all complex numbers are respectively called Integer ring Z 、 Rational number ring Q 、 Real number ring R and Complex ring C ； The addition and multiplication of logarithms form a ring; Even sets are number rings, called Even ring ； And various Algebraic integer ring Only the number set {0} made of the number "zero" is also a number ring.^[1]

Chinese name: Number ring
Foreign name: number ring

Properties: Is the most basic example and model of ring
Discipline: mathematics
Example: Integer ring Z, rational number ring Q, etc

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Basic concepts

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Number ring, yes Number set An algebraic structure of S, which contains at least one number set S, if addition , subtraction, multiplication closure, that is, any two numbers a, b, a+b, a-b, a · b in S are all in S, which is called S composition Number ring.^[2]

Figure 1

If the result of any two numbers in a number set after some operation is still the number in the number set, then the number set is said to be related to this operation close 。

Thus, the number ring is On the Closed Non empty Number Set of Addition, Subtraction and Multiplication 。 The concept of ring in algebra is just the generalization of the concept of number ring.

Example of number ring

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The set consisting of only one number 0, namely {0}, is also a number ring, because 0+0=0, 0-0=0, 0 × 0=0, this number ring is called Zero loop It is the smallest number ring, that is, all other number rings contain it.

If the number ring S contains a non-zero number a, then S must contain an infinite number.

The set of all integers Z is a number ring, because the sum, difference and product of integers are also numbers, this number ring is called Integer ring.

The set of natural numbers is not a number ring, because the difference of natural numbers is not necessarily natural numbers.

For an integer n, the set of all integer multiples of n forms a number ring. In particular, n=2, all even sets form a number ring, called Even ring , recorded as 2Z.

whole Rational number set Q、 All Real Number Sets R、 All complex sets C forms a number ring, which is called Integer ring Z 、 Rational number ring Q 、 Real number ring R and Complex ring C 。 The theorem of division with remainder in integer ring Z is true, and integer theory is just the theory that studies the properties of integer rings.^[2]

All odd sets cannot form a number ring, because the sum of two odd numbers is no longer odd.

The integer set of all forms such as 3n+2 does not form a number ring.

All shapes are like

(m, n are integers) to form a number ring.

nature

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Property 1 Any number ring contains the number zero (that is, the zero ring is the smallest number ring).

Property 2 S Is a number ring, if a ∈ S , then na ∈ S ( n ∈ Z )。

Nature 3 If M 、 N Are number rings, then M ∩ N It is also a number ring.

Analysis of typical examples

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Example 1 Does the four operations of all the logarithms of natural numbers form a ring or field?

Solution: The addition and multiplication of all logarithms of natural numbers do not form a ring or a field.

Because the subtraction of two natural numbers is not necessarily a natural number, the addition, subtraction and multiplication of natural numbers with respect to numbers do not form a ring or a domain.

Example 2 It is proved that if a number ring S ≠ {o), then S contains an infinite number.

prove: set up

, according to the property of number ring, then

, thus

All belong to S, and when

When,

Thus, it is proved that S contains an infinite number.

Example 3 It is proved that the intersection of two number rings is still a number ring; Is the union of two number rings a number ring?

prove: (i) Set

Are two number rings,

, so

And

, thus

, and

, so

It is a number ring.

(ii) The two number rings are not necessarily number rings, for example,

, obviously

, then

, but

, so

It is not a number ring.^[3]

ring

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ring It is an important concept in modern mathematics. Two algebraic operations are specified for a set (usually called addition and multiplication respectively), so that addition satisfies the associative law and commutative law, multiplication satisfies the associative law, and multiplication satisfies the distributive law for addition; There are also zero elements in this set, which is an element that is equal to any element in the set when added to it. Each element has a negative element, and the addition of any element and its negative element is equal to zero: this set is called“ ring ”。 If the multiplication of a ring satisfies the commutative law, it is called a commutative ring. A ring whose elements are numbers is called“ Number ring ”； For example, all integers form a number ring.^[4]