If ring (R1, +, *) is isomorphic to ring (R2, +, *) with f being the isomorphism, then prove the following statements: a. f(e1) = e2, where e1 is the identity element for the group (R1, +) and e2 is the identity element of the group (R2, +). b. f(-a) = -f(a), for each element ‘a’ in R1, where –a represent the additive inverse of element ‘a’ in R1 and –f(a) represents the additive inverse of the element f(a) in R2. c. If (R1, +, *) is a ring with (multiplicative) identity I1 and (R2, +, *) is a ring with (multiplicative) identity I2, then f(I1) = I2.
If ring (R1, +, *) is isomorphic to ring (R2, +, *) with f being the isomorphism, then prove the following statements: a. f(e1) = e2, where e1 is the identity element for the group (R1, +) and e2 is the identity element of the group (R2, +). b. f(-a) = -f(a), for each element ‘a’ in R1, where –a represent the additive inverse of element ‘a’ in R1 and –f(a) represents the additive inverse of the element f(a) in R2. c. If (R1, +, *) is a ring with (multiplicative) identity I1 and (R2, +, *) is a ring with (multiplicative) identity I2, then f(I1) = I2.
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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Question
If ring (R1, +, *) is isomorphic to ring (R2, +, *) with f being the isomorphism,
then prove the following statements:
a. f(e1) = e2, where e1 is the identity element for the group (R1, +) and e2 is
the identity element of the group (R2, +).
b. f(-a) = -f(a), for each element ‘a’ in R1, where –a represent the additive
inverse of element ‘a’ in R1 and –f(a) represents the additive inverse of the
element f(a) in R2.
c. If (R1, +, *) is a ring with (multiplicative) identity I1 and (R2, +, *) is a ring
with (multiplicative) identity I2, then f(I1) = I2.
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