Note 1
Ring
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 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 aN = 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]
Tokyo October 4, 2007
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