Give an example with justification,of a function #f:R_1 rarr R_2#, where #(R_1,+,)# and #(R_2,+,)# are rings and such that #f:(R_1,+) rarr (R_2,+)# is a group homomorphism but #f# is not a ring homomorphism?

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Answer 1 · 2018-02-24T20:12:03

#R_1 = R_2 = ZZ# with #f(n) = 2n#

We can choose #R_1 = R_2 = ZZ# with the normal definitions of #+# and #*# with #f(n) = 2n#.

Then for any #a, b in R_1# we find:

#f(a) + f(b) = 2a + 2b = 2(a+b) = f(a+b)#

So #f# is a group homomorphism.

However:

#f(a)f(b) = (2a)(2b) = 4ab != 2ab = f(ab)#

So #f# is not a ring homomorphism.

Give an example with justification,of a function #f:R_1 rarr R_2#, where #(R_1,+,*)# and #(R_2,+,*)# are rings and such that #f:(R_1,+) rarr (R_2,+)# is a group homomorphism but #f# is not a ring homomorphism?