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1. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx € / for every x,y E R. Deduce that if K, and K₂ are ideals of R and both R/K, and R/K₂ are commutative, then R/(K₁ K₂) is also commutative. 2. Suppose that D is an integral domain and that J and K are ideals of D neither of which equals {0}. Show that Jn K = {0}.

1. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx € / for every x,y E R. Deduce that if K, and K₂ are ideals of R and both R/K, and R/K₂ are commutative, then R/(K₁ K₂) is also commutative. 2. Suppose that D is an integral domain and that J and K are ideals of D neither of which equals {0}. Show that Jn K = {0}.

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 4E: [Type here] Examples 5 and 6 of Section 5.1 showed that is a commutative ring with unity. In...
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Let / be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx e J for every x,y e R. Deduce that if K, and K₂ are ideals of R and both R/K, and R/K₂ are commutative, then R/(K₁ K₂) is also commutative. 1. 2. Suppose that D is an integral domain and that J and K are ideals of D neither of which equals {0}. Show that JK # {0}.
Transcribed Image Text:Let / be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx e J for every x,y e R. Deduce that if K, and K₂ are ideals of R and both R/K, and R/K₂ are commutative, then R/(K₁ K₂) is also commutative. 1. 2. Suppose that D is an integral domain and that J and K are ideals of D neither of which equals {0}. Show that JK # {0}.
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