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AP LANGUAGE and COMPOSITION COURSE DESCRIPTION and SYLLABUS Lakeside High School English Department COURSE DESCRIPTION The purpose of this course is to provide competent, motivated students an opportunity to do  college­level work in high school.  The course organization is aligned with the requirements and  guidelines of the current AP English Course Description and “engages students in becoming skilled  readers of prose written in a variety of periods, disciplines, and rhetorical contexts and in becoming  skilled writers who compose for a variety of purposes.  Both their writing and their reading should  make students aware of the interactions among writer's purposes, audience expectations, and  subjects as well as the way generic conventions and the resources of language contribute to the  effectiveness in writing” (AP English Language and Composition: 2005­2006 Workshop Materials,  49).   LEARNING OBJECTIVES  Upon completing the AP Language and Composition course, students should be able to:  analyze and interpret samples of good writing, identifying and explaining an author's use of  rhetorical strategies and techniques;  apply effective strategies and techniques in their own writing;  create and sustain arguments based on readings, research, and/or personal experience;  demonstrate understanding and mastery of standard written English as well as stylistic  maturity in their own writings;  write in a variety of genres and contexts, both formal and informal, employing appropriate  conventions;  produce expository, analytical, and argumentative compositions that introduce a complex  central idea and develop it with appropriate evidence drawn from source material, cogent  explanations, and clear transitions;   demonstrate an understanding of the conventions of citing primary and secondary source  material;  move effectively through the stages of the writing process with careful attention to inquiry  and research, drafting, revising, editing, and review;  analyze images as text; and  evaluate and incorporate reference documents into research papers (AP English Language   and Composition: 2005­2006 Workshop Materials
  • ap language and composition course description and syllabus lakeside high school english department course description the purpose of this course is to provide competent
  • c.  vocabulary students develop and improve vocabulary by utilizing the sadlier­oxford series
  •  the adventures of huckleberry finn and extensive critical readings viewings
  • spring semester third quarter readings
  • test

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LecturesonCommutativeAlgebra
SudhirR.Ghorpade
IndianInstituteofTechnology Bombay
Annual Foundation School - II
(Sponsored by the National Board for Higher Mathematics)
Bhaskaracharya Pratishthana, Pune
and
Department of Mathematics, University of Pune
June 2006©SudhirR.Ghorpade
DepartmentofMathematics
IndianInstituteofTechnologyBombay
Powai,Mumbai400076,India
E-Mail: srg@math.iitb.ac.in
URL:http://www.math.iitb.ac.in/∼srg/
Version1.1,April28,2008
[Original Version(1.0): June1,2006]
2Contents
1 RingsandModules 4
1.1 IdealsandRadicals . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 PolynomialringsandLocalizationofrings . . . . . . . . . . 9
1.3 Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 ZariskiTopology . . . . . . . . . . . . . . . . . . . . . . . . . 13
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 NoetherianRings 18
2.1 NoetherianRingsandModules . . . . . . . . . . . . . . . . . 18
2.2 PrimaryDecompositionofIdeals . . . . . . . . . . . . . . . . 20
2.3 ArtinianRingsandModules . . . . . . . . . . . . . . . . . . . 24
2.4 Krull’s PrincipalIdealTheorem . . . . . . . . . . . . . . . . . 28
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3 IntegralExtensions 33
3.1 IntegralExtensions . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2 NoetherNormalization . . . . . . . . . . . . . . . . . . . . . . 36
3.3 FinitenessofIntegralClosure . . . . . . . . . . . . . . . . . . 39
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 DedekindDomains 45
4.1 DedekindDomains . . . . . . . . . . . . . . . . . . . . . . . . 46
4.2 ExtensionsofPrimes . . . . . . . . . . . . . . . . . . . . . . . 51
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
A PrimaryDecompositionofModules 56
A.1 AssociatedPrimesofModules . . . . . . . . . . . . . . . . . . 56
A.2 PrimaryDecompositionofModules . . . . . . . . . . . . . . 59
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
References 63
36
Chapter1
RingsandModules
Inthis chapter,we shall reviewa numberofbasic notionsand resultscon-
cerning rings and modules. First, let us settle the basic terminology and
notationthatweshallusethroughoutthesenotes.
By a ring we mean a commutative ring with identity. Given a ring A,
∗ ×we denote byA the set of all nonzero elements ofA and byA the set of
all (multiplicative) units of A. For setsI,J, we writeI ⊆ J to denote that
I is a subset of J and I ⊂ J to denote that I is a proper subset of J, that
is, I ⊆ J and I = J. We denote the set of nonnegative integers byN and
nforanyn∈N,byN wedenotethesetofalln-tuplesofelementsofN. We
sometimesusetheabbreviation‘iff’ tomean‘ifandonlyif’.
1.1 IdealsandRadicals
Historically, the notion of an ideal arose in an attempt to prove Fermat’s
Last Theorem (see Chapter 4 for more on this). From a formal viewpoint,
an ideal of a ring is analogous to a normal subgroup of a group. More
precisely,anidealofaringAisasubsetI ofAsatisfying(i)I isasubgroup
ofAwith respectto addition,and (ii) whenevera∈ I andx∈ A, we have
ax ∈ I. If A is a ring and I is an ideal of A, then we can construct a new
ring, denoted by A/I and called the residue class ring or the quotient ring
obtained from “moding out” A by I. The elements of A/I are the cosets
x+I :={x+a :a∈I}wherexvariesoverA. Additionandmultiplication
inA/I isdefinedby(x+I)+(y+I)= (x+y)+I and(x+I)(y+I)=xy+I.
ThefactthatI isanidealofAensuresthatthisadditionandmultiplication
is well-defined andA/I is a ring with respect to theseoperations. Passing
toA/I fromAhastheeffectofmakingIthenullelement. Wehaveanatural
surjective homomorphismq : A → A/I given by q(x) := x+I for x∈ A.
Thekernelofq ispreciselytheidealI. Conversely,ifφ :A→B isanyring
homomorphism (that is, a map of rings satisfying φ(x+y) = φ(x)+φ(y)
and φ(xy) = φ(x)φ(y) for every x,y ∈ A), then the kernel of φ (which,
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by definition, is the set of all a ∈ A such that φ(a) = 0) is an ideal of A;
moreover, if I = kerφ denotes the kernel of φ, then A/I is isomorphic to
the image of φ. In short, residue class ring and homomorphic image are
identical notions. If I is an ideal of a ring A, then there is a one-to-one
correspondencebetweentheidealsofAcontainingI andtheidealsofA/I
′ −1 ′givenbyJ →q(J) =J/I andJ →q (J ).
An easy way to generate examples of ideals is to look at ideals gen-
erated by a bunch of elements of the ring. Given a ring A and elements
a ,...,a ∈A,theset1 n
(a ,...,a ) :={a x +···+a x :x ,...,x ∈A}1 n 1 1 n n 1 n
isclearlyanidealofAanditiscalledtheidealgeneratedby a ,...,a . More1 n
generally, given any ring A and a subsetE ofA, by EA we denote the set
ofall finiteA-linearcombinations ofelementsofE. Clearly,EAisanideal
of A and it is called the ideal generated by E. Ideals generated by a single
elementare called principal. Thus,anidealI ofaringAiscalled aprincipal
ideal if I = (a) for some a ∈ A. By a principal ideal ring or PIR we mean a
ring in which every ideal is principal. An integral domain which is also a
PIRiscalled aprincipal ideal domain orsimply,aPID.
All the basic algebraic operations are applicable to ideals of a ring. Let
AbearingandletI andJ beidealsofA. ThesumofI andJ isdefinedby
I +J :={a+b : a∈ I, b∈ J}, whereas the product ofI and J is defined
P
by IJ := { a b : a ∈ I, b ∈ J}. Clearly, I +J and IJ are ideals of A.i i i i
ItmayberemarkedthattheproductIJ iscloselyrelated,butnotquitethe
same as, the idealI∩J given by theintersectionofI andJ. Forexample,
ifA is a PID,I = (a) andJ = (b), thenIJ = (ab) whereasI∩J = (ℓ) and
I+J = (d),whereℓ = LCM(a,b)andd = GCD(a,b). Analogueofdivision
is given by the colon ideal (I : J) := {a ∈ A : aJ ⊆ I}. Note that (I : J)
ideal of A. If J equals a principal ideal (x), then (I : J) is often denoted
simply by (I : x). For example, if A is a PID, I = (a) and J = (b), then
(I : J) = (a/d), where d = GCD(a,b). We can also consider the radical of√
nanidealI. It is definedby I :={a∈ A : a ∈ I forsomen≥ 1} and it is
readily seento be anidealofA(by Binomial Theorem!). One saysthatI is√
a radical ideal if I = I. Note that the notions of sum and intersections of
ideals extend easily to arbitrary families of ideals, whereas the notion of a
productofidealsextendseasilytofinitefamilies ofideals.
Having defined algebraic operations for ideals, it is natural to see if
the basic notions of arithmetic find an analogue in the setting of ideals. It
turns out that the notion of a prime number has two distinct analogues as
follows. Let A be a ring and I be an ideal of A. We say that I is prime if
I = A and for anya,b∈ A,, wheneverab∈ I, we have a∈ I orb∈ I. We
saythatI ismaximalifforanyidealJ ofAsatisfyingI⊆J,wehaveJ =I
orJ = A. The set of all prime ideals ofA is denotedby Spec(A), whereas
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thesetofallmaximalidealsofAisdenotedbyMax(A).Itiseasytoseethat
I isaprimeidealifandonlyifA/I isanintegraldomain,andalsothatI is
amaximal idealifandonlyifA/I isafield. Usingthis(oralternatively,by
a simple direct argument), we see that every maximal ideal is prime, that
is,Max(A)⊆ Spec(A).
Examples1.1. (i) IfAisthezeroring,thenSpec(A) =∅ = Max(A).
(ii) IfAisafield,thenSpec(A) ={(0)} = Max(A).
(iii) If A = Z, then Spec(A) = {(0)}∪{(p) : p isaprimenumber}, and
Max(A) ={(p) : p isaprimenumber}.
If A is a ring and P is a nonunit ideal of A, that is, P is an ideal of A
satisfyingP = A, then it is evident thatP is a prime ideal if and only ifP
nsatisfiesthefollowingproperty: if∩ I ⊆P foranyidealsI ,...,I ofA,j 1 nj=1
thenI ⊆ P for somej. It may be interesting to note that there is also thej
following counterpart where instead of an intersectionof ideals contained
inaprimeideal,wehaveanidealcontainedinaunionofprimeideals.
Proposition 1.2 (Prime AvoidanceLemma). Let I, P ,...,P be ideals in a1 n
nringAsuch thatP ,...,P are prime. IfI⊆∪ P ,thenI ⊆P for somej.1 n j jj=1
Proof. Thecasen = 1istrivial. Supposen> 1. Ifthereexistx ∈I\∪ Pi j=i j
for1≤i≤n,thenwehaveacontradictionsincex +x x ...x ∈I\∪ P .1 2 3 n i i
ThusI⊆∪ P ,forsomei. Thecaseofn = 1beingtrivial, theresultnowj=i j
followsusinginductiononn.
Remark1.3. Aneasy alteration of the above proofshows that Proposition
1.2 holdsundertheweakerhypothesisthatI is a subsetofAclosedunder
additionandmultiplication,andP ,...,P areidealsofAsuchthatatleast1 n
n−2 of them are prime. If A contains a field, then Proposition 1.2 can be
proved, by elementary vector space arguments, without assuming any of
theP ’stobeprime.i
Thenotionofcongruencemoduloanintegerhasastraightforwardana-
logueforideals. IfAisaringandI isanidealofA,thenforanyx,y∈Awe
say thatx≡ y(modI) ifx−y ∈ I. More interestingly,Chinese Remainder
Theoremforintegershasthefollowinganalogueforideals.
Proposition 1.4 (Chinese Remainder Theorem). Let I ,I ,...,I be pair-1 2 n
wise comaximal ideals inaringA(i.e.,I +I =Afor alli =j). Then:i j
(i) I I ...I =I ∩I ∩···∩I .1 2 n 1 2 n
(ii) Given any x ,...,x ∈ A, there exists x ∈ A such that x ≡ x (mod I )1 n j j
for1≤j≤n.
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(iii) Themapx+I I ···I →(x+I ,...,x+I ) defines an isomorphism of1 2 n 1 n
A/I I ...I onto thedirect sumA/I ⊕A/I ⊕···⊕A/I .1 2 n 1 2 n
Proof. (i)Clearly,I I ...I ⊆I ∩I ∩···∩I . Toprovetheotherinclusion,1 2 n 1 2 n
we induct on n. The case of n = 1 is trivial. Next, if n = 2, then we can
find a ∈ I and a ∈ I such that a +a = 1. Now, a ∈ I ∩I implies1 1 2 2 1 2 1 2
that a = aa +aa , and thus a ∈ I I . Finally, if n > 2, then as in (i), let1 2 1 2
J = I ···I and note that I +J = A. Hence by induction hypothesis1 2 n 1 1
andthecaseoftwoideals,I ∩I ∩···∩I =I ∩J =I J =I I ···I .1 2 n 1 1 1 1 1 2 n
(ii) Given any i ∈ {1,...,n}, let J = I ···I I ···I . Since I +i 1 i−1 i+1 n i
I = A, we can find a ∈ I such that a ≡ 1(mod I ), for all j = i. Letj ij j ij iQ
a = a . Then a ≡ 1(mod I ) and a ∈ J . Thus I +J = A. Now,i ij i i i i i ij=i
x =x a +···+x a satisfiesx≡x (modI )for1≤j≤n.1 1 n n j j
(iii)Themapx+I I ···I →(x+I ,...,x+I )isclearlywell-defined1 2 n 1 n
andahomomorphism. By(i),itissurjectiveandby(ii),itisinjective.
Of course, not every notion concerning rings is a straightforward ana-
logueofanalgebraic orarithmeticnotionapplicable to integers. Thereare
some basic notions, such as those defined below, which would be quite
redundantoruselessintherealmofintegers.
LetA be a ring. An element a of A is said to be a zerodivisor if there is
∗b∈ A such thatab = 0. We will denotethe set of all zerodivisors inA by
Z(A). Note thatZ(A) ={a ∈ A : (0 : a) = 0}. An element of A which is
not a zerodivisor is called a nonzerodivisor (sometimes abbreviated as nzd).p
nElementsof (0), that is, thoseelementsa∈ A forwhicha = 0 forsomep
n ∈ N, are said to be nilpotent. The set (0) of all nilpotent elements in a
ringAiscalled thenilradical ofA. IfAisanonzeroring,thenclearly everyp
nilpotentelementofAisazerodivisor,thatis, (0)⊆Z(A).
Interestingly,each ofthe above notionsis neatly connectedwith prime
ideals. To begin with, the set of all nonzerodivisors in a ring enjoys prop-
ertiessimilar tocomplementsofprime ideals. Moreprecisely,it is a multi-
plicatively closedsetinthefollowingsense.
Definition1.5. AsubsetS ofaringAissaidtobemultiplicatively closedifit
satisfiesthefollowing: (i)1∈S and(ii)ifa∈S andb∈S,thenab∈S.
∗Examples1.6. (i) If A is an integral domain, thenA = A\{0} is mul-
tiplicatively closed. More generally,as remarkedearlier,thesetofall
nonzerodivisorsinanyringAisamultiplicativelyclosedsubsetofA.
×Also, the setA of multiplicative units in any ring A is a multiplica-
tivelyclosedsubsetofA.
(ii) If P is a prime ideal of a ring A, then A\P multiplicatively closed.
Moregenerally,if{P :α∈ Λ}isafamily ofprimeidealsofaringA,α
thenA\∪ P is a multiplicatively closedsubsetofA. Notethat ifα∈Λ α
7I is any ideal in a ring A, thenA\I is multiplicatively closed if and
onlyifIisaprimeideal.
n(iii) Given any element a of a ring A, the set S = {a : n ∈ N} of pow-
ers of a is a multiplicatively closed subset of A. In particular,{1} is
multiplicatively closedanditisclearlythesmallestamongthemulti-
plicatively closedsubsetsofA.
(iv) If a ring A is a subring of a ring B and S is a multiplicatively closed
subsetofA,thenS isamultiplicatively closedsubsetofB.
Lemma1.7. LetAbearing. IfI isanidealofAandS isamultiplicativelyclosed
subset ofA such that I∩S =∅, then there exists a prime ideal P ofA such that
I ⊆P andP ∩S =∅. MoreoverP is maximal amongthe family of idealsJ ofA
satisfyingI⊆J andJ∩S =∅.
Proof. Considerthe family{J : J anidealofAwithI ⊆ J andJ∩S =∅}
andZornify!
Corollary1.8. LetI beanonunitideal ofaringA. Thenthere isamaximalideal
mofAsuchthatI⊆m. Inparticular, everynonzeroringhasamaximalidealand
the spectrum ofanonzero ring isnonempty.
Proof. The first assertion follows from Lemma 1.7 withS ={1}. To prove
thesecondassertion,takeI = (0).
Corollary1.9. LetAbearingandI beanyideal ofA. Then
√ \ p \
I = P. Inparticular, (0) = P.
P∈ Spec(A) P∈ Spec(A)
I⊆P

Proof. Clearly I ⊆ P for every prime ideal P of A containing I. On the√
other hand, if a∈ P for everyP ∈ Spec(A) withI ⊆ P, buta ∈ I, then
napplyingLemma1.7toS :={a :n∈N},wearriveatacontradiction.
nIfa is a nilpotent element of a ringA thena = 0 for somen∈N, and
2 n−1hence(1−a)(1+a+a +···+a ) = 1. Inotherwords,wehaveavalid
geometricseriesexpansion
1 n−1= 1+a+···+a ,
1−a
which shows that 1−a is a unit in A. In fact, a similar argument shows
that if a ∈ A is nilpotent, then 1− ab is a unit in A for any b ∈ A. Itp
follows that (0) ⊆ {a ∈ A : 1− abisaunitforeveryb ∈ A}. The set
{a ∈ A : 1− abisaunitforeveryb ∈ A} will be denoted by J(A) and
calledtheJacobsonradicalofA. ThefollowingresultshowsthattheJacobson
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radical of A is an an ideal of A and it is, in fact, the intersection of all the
maximal ideals of A. It also gives an alternative proof of the fact that the
nilradical iscontainedintheJacobsonradical.
Proposition1.10. LetAbe aring. Then
\
J(A) = m
m∈Max(A)
Proof. Let a ∈ J(A). If a ∈/ m for some m ∈ Max(A), then m + (a) is an
ideal of A with m ⊂ m + (a). Since m is maximal, we have m + (a) = A.
Hencethereism∈ mandb∈Asuchthatm+ab = 1.Nowmcontainsthe
unitm = 1−ab, and thisis a contradiction. Ontheotherhand,ifais inm
foreverym∈ Max(A), thensoisabforeveryb∈A. Nowif 1−abisnota
unit inA forsomeb∈ A, thenby Corollary 1.8, thereism∈ Max(A) such
that1−ab∈m,butthen1∈m,whichisacontradiction.
1.2 PolynomialringsandLocalizationofrings
Formingtheresidueclassringorthequotientringisoneofthethreefunda-
mental processesin Algebra for constructing new rings from a given ring.
Theothertwoprocessesare forming thepolynomial ringin oneorseveral
variableswithcoefficientsinthegivenringandformingthelocalization of
thegivenring. Weshallfirstreviewtheformerandthendescribethelatter.
Polynomial Ring: Let A be a ring and n be a nonnegative integer.
We denote by A[X ,...,X ] the ring of all polynomials in the variables1 n
X ,...,X withcoefficientsinA. ElementsofA[X ,...,X ]looklike1 n 1 n
X
i i1 nf = a X ...X , a ∈A,i ...i i ...i1 n 1 n 1 n
nwhere(i ,...,i )varyoverafinitesubsetofN . Atypicalterm(excluding1 n
i i1 nthe coefficient), viz., X ...X , is called a monomial; its (usual) degree is1 n
i +···+i . Suchamonomialissaidtobesquarefree ifi ≤ 1for1≤r≤n.1 n r
Iff = 0,thenthe(total)degreeoff isdefinedbydegf = max{i +···+i :1 n
a = 0}. Usual convention is that deg0 = −∞. With this in view, fori ...i1 n
every f,g ∈ A[X ,...,X ], we have deg(f +g) ≤ max{degf,degg} and1 n
in case A is a domain, then degfg = degf +degg. A homogeneous polyno-
mialofdegreedinA[X ,...,X ]issimply afiniteA-linear combination of1 n
monomialsofdegreed. Thesetofall homogeneouspolynomialsofdegree
d is denoted by A[X ,...,X ] . Note that any f ∈ A[X ,...,X ] can be1 n 1 nd
uniquelywrittenasf =f +f +...,wheref ∈A[X ,...,X ] andf = 00 1 i 1 n ii
fori> degf;wemaycallf ’stobethehomogeneouscomponentsoff. Iff = 0i
andd = degf,thenclearlyf = 0andf =f +f +···+f . AnidealI ofd 0 1 d
A[X ,...,X ] is said to be a homogeneous ideal (resp: monomial ideal) if it is1 n
9generated by homogeneous polynomials (resp: monomials). Henceforth,
when we use a notation such as k[X ,...,X ], it will be tacitly assumed1 n
thatk denotesafieldandX ,...,X areindependentindeterminatesover1 n
k (and,ofcourse,n∈N).
The process of localization described next generalizes the construction
of the field of fractions of an integral domain, which in turn, is a general-
izationoftheformalconstructionofrationalnumbersfromintegers.
Localization: LetAbe aring andS be amultiplicatively closed subset
ofA. Definearelation∼onA×Sasfollows. Givenany(a,s),(b,t)∈A×S,
(a,s)∼ (b,t) ⇐⇒ u(at−bs) = 0forsomeu∈S.
It is easy to see that ∼ defines an equivalence relation on A× S. Let us
a −1denotetheequivalenceclassof(a,s)∈A×S bya/s(orby ),andletS A
s
denotethesetofequivalenceclassesofelementsofA×S. Defineaddition
−1andmultiplicationonS Aby
a b at+bs a b ab
+ = and = forany(a,s),(b,t)∈A×S.
s t tb s t ts
It can be easily seenthat thesebinary operationsare well definedand that
−1 −1S A is a ring with respect to them. The ring S A is called the ring of
fractionsorthelocalizationofAwithrespecttomultiplicativelyclosedsubset
−1S. PassingtoS AfromAhastheeffectofmakingtheelementsofS units.
−1 −1Incase 0∈ S, we seethatS A is the zero ring,and conversely,ifS A is
thezeroring,then0∈S.
Examples1.11. LetAbearing.
−1(i) IfAisanintegraldomainandS =A\{0}, thenS Aisnothingbut
the quotient field or the field of fractions ofA. Note that in this case
the equivalence relation∼ onA×S takes the simpler form: (a,s)∼
(b,t) ⇐⇒ (at−bs) = 0. More generally, this is the case when all
theelementsofS are nonzerodivisors. Ingeneral,ifS isthesetofall
−1nonzerodivisorsinA,thenS Aiscalled thetotal quotientringofA.
−1(ii) Let S = A\ p, where p is a prime ideal of A. In this case S A is
customarily denoted by A . The set pA := {a/s : a ∈ p, s ∈ S} isp p
an ideal ofA and an element ofA that is not inpA is a unit inA .p p p p
It follows that pA is the only maximal ideal of the ring A . In otherp p
wordsA is a local ring [A local ring is a ring withonly one maximalp
ideal].
Ingeneral,wehavethenaturalhomomorphism
a−1φ :A→S A definedby φ(a) := fora∈A.
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