Distance Theory Algebraically Supplemented Note 1 Ring / October 4, 2007

Distance Theory Algebraically Supplemented

Note 1

Ring

TANAKA Akio

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<Ring>     To be meant commutative ring containing 1 and defining the operation addition and multiplication

<Ring of rational integers>     Set of all the integers containing the operation addition and multiplication

<Ring of polynomials in n variables over k>     Ring that is what k is field x₁, …, x_n is variables and k-coefficient n-variables all the polynomials set has addition and multiplication.

<Zero divisor>     When commutative ring R that has element a , there exists element b ≠ 0 in the condition ab = 0. a is zero divisor.

<Integral domain>     Ring that has not zero divisor except 0

<Ideal of R>    Subset I that satisfies next conditions in R  

(1) a, b ∈ I ⇒ = -a +b ∈ I

(2) a ∈ I, r ∈ R ⇒ ra ∈ I

Ideal defines addition, multiplication and quotient.

<Prime ideal of R>     Ring R’s ideal p≠R     a, b∈R, ab∈p ⇒ a∈p or b∈p

<Maximal ideal of R>     Ring R’s ideal p≠R     m ⊂_≠ a ⊂_≠ R     When there does not exist ideal a, m is maximal ideal.

<Principal ideal>     Ideal generated by ring R’s one element a   (a) = { xa | x=R }

<Principal ideal ring>     R’s all the ideal are principal ideals

<Principal ideal domain>     Principal ideal ring when R is integral domain

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<Radical> of a     Commutative ring R’s ideal a is in the condition √a = { a ∈ R | Natural number n has aⁿ ∈ a }.

<Nilradical>       √(0)

<Primary ideal>     R is ring. q ( ≠R ) is ideal of R.     When a, b ∈ R, ab ∈ q and a ∉ q, there exists natural number n and bⁿ ∈ q.

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<Multiplicatively closed set>

R is ring. S is subset of R. When S has next conditions, S is multiplicatively closed set.

(1) x, y∈S   xy∈S

(2) 1∈S

(3) 0 ∉ S

<Quotient ring>

Class containing elements (r, s) is expressed by r/s.

When S^-1R (set of class r/s) has next conditions, S^-1R is quotient ring. Quotient ring has unit 1/1 and zero element is 0/1.

(1) (a₁/s₁) + (a₂/s₂) = (a₁s₂ + a₂s₁)/s₁s₂

(2) (a₁/s₁)(a₂/s₂) = a₁a₂/s₁s₂

<Ring of total quotients>

R is ring. S is all the non-zero divisors. q( R ) = S^-1R is ring of total quotient.

<Quotient field>

q( R ) that has inverse elements except element 0

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<Local ring>

Commutative ring that has only one maximal ring

<Residue field>

R is local ring. m is R’s maximal ideal. R/m is residue field of R.

<Localization>

R’s prime ideal is p. S = R∖p is multiplicatively closed set.

S = R∖p    Rp = S^-1R    Rp is localization of R at p.

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<Noetherian ring>

Noetherian ring satisfies next conditions.

(1) Commutative ring R has maximal one in arbitrary set that ideals of R make.

(2) Infinite sequence of R has number N that is a_N = a _N+1 = … .

(3) Arbitrary ideal of R is finitely generated.

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<Hilbert basis theorem>

When R is Noetherian ring, ring of polynomials in n-variables over R is also Noetherian ring.

[Reference]

Algebraic Linguistics / Linguistic Result / Deep Fissure between Word and Sentence / Tokyo September 10, 2007

Tokyo October 4, 2007

Sekinan Research Field of Language