Number ring is a special number set. The ring composed of numbers isringThe 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 aNumber 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 calledInteger ring Z、Rational number ring Q、Real number ring RandComplex ring C;The addition and multiplication of logarithms form a ring;Even sets are number rings, calledEven ring;And variousAlgebraic integer ringOnly the number set {0} made of the number "zero" is also a number ring.[1]
Number ring, yesNumber setAn algebraic structure of S, which contains at least one number set S, ifaddition, 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 compositionNumber 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 operationclose。
Thus, the number ring isOn 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 calledZero loopIt 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 calledInteger 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, calledEven ring, recorded as 2Z.
wholeRational number setQ、All Real Number SetsR、All complex setsC forms a number ring, which is calledInteger ring Z、Rational number ring Q、Real number ring RandComplex 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 2SIs a number ring, ifa∈S, thenna∈S(n∈Z)。
Nature 3 IfM、NAre number rings, thenM∩NIt is also a number ring.
Analysis of typical examples
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Example 1Does 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 2It 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, thusAll belong to S, and whenWhen,Thus, it is proved that S contains an infinite number.
Example 3It 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) SetAre two number rings,, soAnd, thus, and, so, soIt is a number ring.
(ii) The two number rings are not necessarily number rings, for example,, obviously, then, but, soIt is not a number ring.[3]
ring
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ringIt 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]
On Number Fields
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definitionLet F be a number ring, if
(i) F contains a number that is not equal to zero;