1. Let R be a commutative ring and a any element in R. Define the annhilator of a to be the set ann(a) = {r € R | ra = 0}(that is, the set of all elements that multiply ato zero). Prove that ann(a)is an ideal of R. 2. Referring to problem 1, we will calculate the annihilators of elements in various rings: (a) Let R = Z6 (the integers modulo 6) - determin ann([2]) and ann([5]) (that is, the annihilators of the classes [2] and 5]) (b) Let R Z18 (the integers modulo 18) - determine ann([6]) (express it as a principal ideal generated by an element in the fing) = (c) Let R = Z× Z - determine ann((1,0)) (that is, the annihiltor of the ordered pair (1,0))
1. Let R be a commutative ring and a any element in R. Define the annhilator of a to be the set ann(a) = {r € R | ra = 0}(that is, the set of all elements that multiply ato zero). Prove that ann(a)is an ideal of R. 2. Referring to problem 1, we will calculate the annihilators of elements in various rings: (a) Let R = Z6 (the integers modulo 6) - determin ann([2]) and ann([5]) (that is, the annihilators of the classes [2] and 5]) (b) Let R Z18 (the integers modulo 18) - determine ann([6]) (express it as a principal ideal generated by an element in the fing) = (c) Let R = Z× Z - determine ann((1,0)) (that is, the annihiltor of the ordered pair (1,0))
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Can you just answer #2
![1. Let R be a commutative ring and a any element in R. Define the annhilator
of a to be the set ann(a) = {r € R | ra = 0}(that is, the set of all elements
that multiply ato zero). Prove that ann(a)is an ideal of R.
2. Referring to problem 1, we will calculate the annihilators of elements in
various rings:
(a) Let R = Z6 (the integers modulo 6) - determin ann([2]) and ann([5])
(that is, the annihilators of the classes [2] and 5])
(b) Let R
Z18 (the integers modulo 18) - determine ann([6]) (express
it as a principal ideal generated by an element in the fing)
=
(c) Let R = Z × Z - determine ann((1,0)) (that is, the annihiltor of the
ordered pair (1,0))](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32f77ee0-291c-46d0-b315-80fb2fd096d8%2Ffea51a2d-a3af-44f1-9884-0ff1d744f87d%2F2wl1lrf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. Let R be a commutative ring and a any element in R. Define the annhilator
of a to be the set ann(a) = {r € R | ra = 0}(that is, the set of all elements
that multiply ato zero). Prove that ann(a)is an ideal of R.
2. Referring to problem 1, we will calculate the annihilators of elements in
various rings:
(a) Let R = Z6 (the integers modulo 6) - determin ann([2]) and ann([5])
(that is, the annihilators of the classes [2] and 5])
(b) Let R
Z18 (the integers modulo 18) - determine ann([6]) (express
it as a principal ideal generated by an element in the fing)
=
(c) Let R = Z × Z - determine ann((1,0)) (that is, the annihiltor of the
ordered pair (1,0))
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