RingThe definition of is similar to the interchangeable group, but adds another operation "·" on the basis of the original "+" (note
+ And. It is not generally known as addition and multiplication ). In abstract algebra, ResearchRingIsRing Theory.
Definition
The Set R and binary operations defined on it + and (r, +, ·) constituteRingIf they meet the following requirements:
- (R, +) forms an exchange group. Its unit is calledZero Element, As '0 '. That is:
- (R, +) is closed
- (A + B) = (B +)
- (A + B) + C = a + (B + C)
- 0 + A = a + 0 =
- When a minus (−a) satisfies a + −a = −a + A = 0
- (R, ·) forms a semi-group, namely:
- (A · B) · c = A · (B · C)
- (R, ·) is closed
- Multiplication:
- A · (B + C) = (A · B) + (A · C)
- (A + B) · c = (A · c) + (B · C)
Multiplication operator is often omitted, so a. B can be abbreviated as AB. In addition, multiplication is the prior operation of the bitwise addition method, so a + BC is actually a + (B · C ).
Basic Nature
Considering a ring R, according to the definition of the ring, it is easy to know that R has the following properties:
- When a ε r, A · 0 = 0 · A = 0; (this is why 0 is called "zero element" as the unit element of the addition group ")
- When a, B, R, (-a) · B = A · (-B) =-(A · B );
Special Ring
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Ring
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In ring R, (R, ·) forms a semi-group. That is: when 1 is R, so that when a is R, 1 · A = A · 1 =. R is called
Ring. In this case, the inner 1 of the inner group (R, ·) is also called the inner element of the ring R.
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Exchange Ring
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If (R, ·) in the ring R still satisfies the exchange law and forms an exchange group, I .e., a, B, R, and AB = BA, R is called
Exchange Ring.
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No zero-factor ring
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If there is no non-zero factor in R, R is called
No zero-factor ring.
- This definition is equivalent to any of the following:
- R \ {0} forms a semi-group for multiplication;
- R \ {0} is closed to multiplication;
- The product of non-zero elements in R is not 0;
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Integral ring
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An exchange ring without zero factor is called
Integral ring.
Example: Integer Ring and polynomial ring
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Unique decomposition Ring
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If each non-zero non-reversible element in the entire ring R can be uniquely decomposed, R is called
Unique decomposition Ring.
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Division Ring
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If the ring R is a ring, and r \ {0} forms a group for multiplication on R, that is, running a ε r \ {0 }, then a ^ {-1} in r \ {0} makes a ^ {-1} · A = A · A ^ {-1} = 1. R is called
Division Ring.
- The division ring is not necessarily an exchange ring. Inverse example: a quaternary ring.
- The switching ring is a body.
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Main ideal ring
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Each ideal is the main ideal integral ring called
Main ideal ring.
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Single ring
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If the great ideal in ring R is zero, R is called
Single ring.
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Shang Huan
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Quality Ring
Example
- Ring: The R of a non-empty set forms a ring. if and only if it meets any of the following conditions:
- R is closed to the sum and Difference Operation of the set, that is, E, F, R, E, F, R, E-F, R;
- R is closed to the intersection and symmetry Difference Operation of the set, that is, E, F, R, E, F, R, E, F, and R;
- R refers to the intersection, difference, and non-intersection operation of the Set.
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In this way, the set ring is orthogonal multiplication, and the symmetry difference is addition. The empty set is zero RMB, therefore, it is a bucket.
- An integer ring is a typical exchange with a unit ring.
- Rational Number ring, real number field, and complex number field are all exchanged meta-rings.
- The coefficients of all items constitute the polynomial of a ring. All a [x] is a ring. It is called a polynomial ring on.
- N is a positive integer, and all the real number matrices of n × n constitute a ring.
