1. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx 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. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if xy-yx 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
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter6: More On Rings
Section6.3: The Characteristic Of A Ring
Problem 4E
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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 € 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.
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}.
3. Let R be the set of all matrices with rational entries, M3(Q), of the form
a b c
0 ab
00 a
Show that
(i) R is a commutative ring.
(ii) the set
is an ideal of R.
(iii) A is a maximal ideal of R.
0bc
--{]^²}
00 b
000](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6fe9af21-d25b-4540-a3cc-0ad469a8dc5b%2F9932ca9a-8689-4786-9ffa-0ce6fe8507ee%2Fhb6lw3_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let ./ be an ideal of the ring R. Show that the ring R/J is commutative if and only if
xy-yx € 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.
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}.
3. Let R be the set of all matrices with rational entries, M3(Q), of the form
a b c
0 ab
00 a
Show that
(i) R is a commutative ring.
(ii) the set
is an ideal of R.
(iii) A is a maximal ideal of R.
0bc
--{]^²}
00 b
000
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