A ring consists of a set R and two binary operations+and+,which can be called addition and multiplication. Rings must comply with the following laws:

(r,+) forms a commutative group, and its unit element is called zero element, which is denoted as' 0'. Namely:

(a + b) = (b + a)

(a + b) + c = a + (b + c)

0 + a = a + 0 = a

Answer? (? A) meet a+? a =？ a + a = 0

(r,) observe:

1 a = a 1 = a 0 = A (including rings only)

a b c = a b c

Multiplication satisfies the distribution law of addition;

a (b + c) = (a b) + (a c)

(a + b) c = (a c) + (b c)

Note that in multiplication, A is often omitted, so ab can be abbreviated as AB. In addition, multiplication takes precedence over addition, so a+bc is actually A+(B C).

Some special rings

Rings containing unit elements:

In the definition of rings, the existence of multiplication unit (1) is not explicitly required. If ring R has a unit for multiplication (called unitary element or unitary element or unitary element, called'1'), it is called unitary ring or unitary ring.

Exchange rings:

Although the definition of a ring requires addition to have an commutative law, it does not require multiplication to have an commutative law. If the multiplication defined above is commutative: ab=ba, then this ring is called commutative ring.

Split ring:

Main entrance: split ring

If a ring containing the identity r is divided by a single bit 0 about addition to form a group for multiplication (generally speaking, the ring r forms a semigroup for multiplication), then the ring is called a division ring. Division rings are not necessarily commutative rings, such as quaternion rings. A commutative division ring is a domain.

Zero-factor-free ring:

Generally speaking, the ring r forms a multiplicative semigroup, but R\{0} does not necessarily form a multiplicative semigroup. Because if the product of two non-zero elements is zero, R\{0} is not a closed multiplication. If R\{0} still constitutes a multiplicative semigroup, then this ring is called a zero-factor-free ring.

This definition is actually equivalent to that the product of any two non-zero elements is non-zero.

The whole ring:

Main items: the whole ring

An integral ring is an commutative ring containing unit elements without zero factors, such as polynomial rings and integer rings.

Principal ideal ring:

Principal term: principal ideal ring

An integral ring in which every ideal is a principal ideal is called a principal ideal ring.

Unique decomposition ring:

Main item: unique decomposition ring

If every non-zero irreversible element in the whole ring R can be uniquely decomposed, then R is called a unique decomposition ring.

Business ring:

Subject: quotient ring

Main ring:

Main products: prime ring

Example:

Integer rings are typical commutative rings with unit rings.

Rational number ring, real number field and complex number field are commutative rings containing identity elements.

The polynomial group A[X] whose coefficients of all terms form a ring is a ring, which is called polynomial ring on a. 。

N is a positive integer, and all real matrices of n×n form a ring.

Ideals of rings

Main project: ideal

Right ideal: Let R be a ring, then the ring R and its addition+form an Abel group. Let I be a subset of R, then I is called the right ideal of R, if the following conditions hold:

(i,+) constitutes a subgroup of (r,+).

For everyone.

Left Ideal: Similarly, I is called the left ideal of R if the following conditions hold:

(i,+) constitutes a subgroup of (r,+).

For everyone.

If I is both a right ideal and a left ideal, I is called a bilateral ideal.

Example:

Ideal of integer ring: integer ring z has only nZ ideal.

Ideals of division rings: Only (left or right) ideals in division rings are ordinary (left or right) ideals.

General nature:

Theorem 1 In a ring, the sum of (left or right) ideals is still (left or right) ideals.

Theorem 2 In a ring, the intersection of (left or right) ideals is still (left or right) ideals.

For the two ideals of R, A and B, remember. By definition, it is not difficult to prove the following basic properties:

(1) If A is the left ideal of R, then AB is the left ideal of R;

(2) If B is the right ideal of R, then AB is the right ideal of R;

(3) If A is R's left ideal and B is R's right ideal, then AB is R's bilateral ideal.

If a nonempty subset of ring R of A makes =RA+AR+RAR+ZA, it is the ideal of ring R. This ideal is called the ideal generated by A in R, and A is called the generating set. The generating subgroup of the same group is similar to the intersection of all ideals containing a in R, so it is the smallest ideal containing a in R. The following are some special cases of generating ideals:

(1) When it is a commutative ring, =RA+ZA.

(2) When it is a ring with a unit of 1, =RAR.

(3) When there is a unit exchange ring, =RA

Principal ideal: if it is an n-tuple set, it is called finite generating ideal. Especially if it is a single element set, it is called the principal ideal of ring R. Note that as a generator, it is generally not unique. For example, in the general form:

Nature:

Principal ideals in some special rings;

(1) If it is a switched ring, then

(2) If it is an identity ring, then

(3) If it is an commutative ring with unit elements, then

Ideal: If I is the proper subset of R, I is called the ideal of R. 。

Maximal ideal: A true ideal I is called the maximal ideal of R. If there is no other true ideal J, I is the proper subset of J. 。

Maximal left ideal: let I be the left ideal of ring R. If there is no true left ideal between I and R, I will be called the maximal left ideal of ring R. The relationship between maximal left ideal and maximal ideal is as follows:

(1) If I is a maximal left ideal and a bilateral ideal, then I is a maximal ideal.

(2) The maximal ideal is not necessarily the maximal left ideal.

The zero ideal of a division ring is a maximal ideal. In a ring with identity, if the zero ideal is its maximum ideal, it is called a simple ring. Division rings are simple rings, and the domain is also simple rings. Otherwise, it is not true, that is, there is a simple ring that is not a division ring.

Theorem 1 In an integer ring Z, the principal ideal generated by P is a maximal ideal if and only if P is a prime number.

Theorem 2 Let R be a commutative ring with the identity 1 The ideal I is the maximal ideal of R if and only if the quotient ring R/I is a domain.

Theorem 3 If I is the left ideal of a ring R, then I is the maximal left ideal of R if and only if I+J = R is not included in I for any left ideal J of R. 。

Prime ideal: the true ideal I is called the prime ideal of R. If any ideal A or B of R is involved, we can deduce or.

Prime ring: If the zero ideal of ring R is prime, it is called prime ring (or prime ring). A ring without a zero factor is a prime ring. In commutative ring R, R/I is a necessary and sufficient condition of prime ring.

Semi-prime ideal: Let I be the ideal of ring R, and if any ideal p can be obtained by, I is said to be the semi-prime ideal of ring R. 。

Obviously, semi-prime ideal is an ideal with relatively weak conditions, because prime ideal is semi-prime ideal, but semi-prime ideal is not necessarily prime ideal.