Wednesday 24 July 2013

Distance Theory Algebraically Supplemented Note 1 Ring

Note 1

<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, …, xn is variables and k-coefficient n-variables all the polynomials set has addition and multiplication.
<Zero divisor>     When commutative ring 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) ab  I  = -a +b  I
(2) a  Ir  R  ra ∈ I
Ideal defines addition, multiplication and quotient.
<Prime ideal of R>     Ring R’s ideal pR     abRab ap or bp
<Maximal ideal of R>     Ring R’s ideal pR     m  a  R     When there does not exist ideal ais maximal ideal.
<Principal ideal>     Ideal generated by ring R’s one element a   (a) = { xa x=}
<Principal ideal ring>     R’s all the ideal are principal ideals
<Principal ideal domain>     Principal ideal ring when R is integral domain
<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. ) is ideal of R.     When a R, ab  and a  q, there exists natural number n and bn  q.
<Multiplicatively closed set>
R is ring. is subset of R. When has next conditions, S is multiplicatively closed set.
(1) xyS   xyS
(2) 1S
(3) 0  S
<Quotient ring>
Class containing elements (rs) is expressed by r/s.
When S-1R (set of class r/s) has next conditions, S-1is 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. qR ) = S-1R is ring of total quotient.
<Quotient field>
qR ) that has inverse elements except element 0
<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.
R’s prime ideal is pRp is multiplicatively closed set.
Rp    Rp = S-1R    Ris localization of R at p.
<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 aN = N+1 = … .
(3) Arbitrary ideal of R is finitely generated.
<Hilbert basis theorem>
When R is Noetherian ring, ring of polynomials in n-variables over R is also Noetherian ring.


Tokyo October 4, 2007

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