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2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a,b) : a € A and be B}. This can be extended naturally to a finite number of sets A1, A2,..., Aµ: Aj x Az x ...x Ak = {(a1,a2,.…..,az) : a; € A; for 1

2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a,b) : a € A and be B}. This can be extended naturally to a finite number of sets A1, A2,..., Aµ: Aj x Az x ...x Ak = {(a1,a2,.…..,az) : a; € A; for 1

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.2: Integral Domains And Fields
Problem 7E: [Type here] 7. Let be the set of all ordered pairs of integers and . Equality, addition, and...
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2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a,b) : a E A and be B}. This can be extended naturally to a finite number of sets A1, A2,..., Ak: Aj × A2 x --x Ak = {(a1,a2, ...,ar) :a; € A; for 1< j < k}. This can be further extended to an infinite number of sets. Let I be an index set and X = {X;}ie1 be a family of sets indexed by the set I. Note that I may be an uncountable set. The Cartesian product of the family X is given by II x. = {f :1¬UX, : f() e X, for all i 1 ię! That is, the elements of the Cartesian product are functions whose domain is the index set I and the image of i is in X; for any i e I. Let {G;}ie1 be a family of groups indexed by the set I. Let e; be the identity element of G¡, for all i e I. Consider the Cartesian product G = IIke, Gi- %3D b. Let W C G be the set consisting of elements g e G such that g(i) = e; for all but finitely many i e I. Show that W 4 G. We call W the weak direct product of the family {G;}ie1. If all groups are abelian, we use an additive notation and call the weak direct product, the external direct sum and W is usually written as Oiei Gi.
Transcribed Image Text:2. Recall that for two sets A, B, the Cartesian product of A and B is the set Ax B = {(a,b) : a E A and be B}. This can be extended naturally to a finite number of sets A1, A2,..., Ak: Aj × A2 x --x Ak = {(a1,a2, ...,ar) :a; € A; for 1< j < k}. This can be further extended to an infinite number of sets. Let I be an index set and X = {X;}ie1 be a family of sets indexed by the set I. Note that I may be an uncountable set. The Cartesian product of the family X is given by II x. = {f :1¬UX, : f() e X, for all i 1 ię! That is, the elements of the Cartesian product are functions whose domain is the index set I and the image of i is in X; for any i e I. Let {G;}ie1 be a family of groups indexed by the set I. Let e; be the identity element of G¡, for all i e I. Consider the Cartesian product G = IIke, Gi- %3D b. Let W C G be the set consisting of elements g e G such that g(i) = e; for all but finitely many i e I. Show that W 4 G. We call W the weak direct product of the family {G;}ie1. If all groups are abelian, we use an additive notation and call the weak direct product, the external direct sum and W is usually written as Oiei Gi.
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