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If I is an ideal of R, then by definition, (I, +) is an abelian group. Consequently, it has an identity element, call it 01, that satisfies the I. property that i+01 = 01+i = i for all i E I. On the other hand, the element "0" in R is the identity element for the group (R,+). Prove that the element 0, must be the same as the element 0.

If I is an ideal of R, then by definition, (I, +) is an abelian group. Consequently, it has an identity element, call it 01, that satisfies the I. property that i+01 = 01+i = i for all i E I. On the other hand, the element "0" in R is the identity element for the group (R,+). Prove that the element 0, must be the same as the element 0.

Advanced Engineering Mathematics
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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absract algebra

Exercise 2.65.1 If I is an ideal of R, then by definition, (I,+) is an abelian group. Consequently, it has an identity element, call it 0r, that satisfies the property that i +01 = 0, + i = i for all i E I. On the other hand, the element "0" in R is the identity element for the group (R, +). Prove that the element 0, must be the same as the element 0. (See Exercise 2.29 of this chapter for some clues if you need. Both this exercise and Exercise 2.29 of this chapter are just special cases of Exercise 3.5 in Chapter 4 ahead.)
Transcribed Image Text:Exercise 2.65.1 If I is an ideal of R, then by definition, (I,+) is an abelian group. Consequently, it has an identity element, call it 0r, that satisfies the property that i +01 = 0, + i = i for all i E I. On the other hand, the element "0" in R is the identity element for the group (R, +). Prove that the element 0, must be the same as the element 0. (See Exercise 2.29 of this chapter for some clues if you need. Both this exercise and Exercise 2.29 of this chapter are just special cases of Exercise 3.5 in Chapter 4 ahead.)
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