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I am reading An Introduction to Rings and Modules With K-Theory in View by A.J. Berrick and M.E. Keating (B&K).
I need help to clarify a remark of B&K regarding ring homomorphisms from the zero or trivial ring ...
The relevant text from B&K reads as follows:
https://www.physicsforums.com/attachments/6078
https://www.physicsforums.com/attachments/6079In the above text from B&K's book we read ...
"... ... This follows from the observation that the obvious map from the zero ring to $$R$$ is not a ring homomorphism (unless $$R$$ itself happens to be $$0$$). ... ... "
I do not understand the above statement that the obvious map from the zero ring to $$R$$ is not a ring homomorphism (unless $$R$$ itself happens to be $$0$$) ... ... What, indeed do B&K mean by the obvious map from the zero ring to $$R$$ ... ... ?It seems to me that the obvious map is a homomorphism ... ..Consider the rings $$T, R$$ where $$T$$ is the zero ring and $$R$$ is any arbitrary ring ... so $$T = \{ 0 \}$$ where $$0 = 1$$ ...
Then to me it seems that the "obvious" map is $$f( 0_T) = 0_R$$ ... ... which seems to me to be a ring homomorphism ...
... BUT ... this must be wrong ... but why ... ?
Can someone please clarify the above for me ...
Some help will be very much appreciated ...
Peter
I need help to clarify a remark of B&K regarding ring homomorphisms from the zero or trivial ring ...
The relevant text from B&K reads as follows:
https://www.physicsforums.com/attachments/6078
https://www.physicsforums.com/attachments/6079In the above text from B&K's book we read ...
"... ... This follows from the observation that the obvious map from the zero ring to $$R$$ is not a ring homomorphism (unless $$R$$ itself happens to be $$0$$). ... ... "
I do not understand the above statement that the obvious map from the zero ring to $$R$$ is not a ring homomorphism (unless $$R$$ itself happens to be $$0$$) ... ... What, indeed do B&K mean by the obvious map from the zero ring to $$R$$ ... ... ?It seems to me that the obvious map is a homomorphism ... ..Consider the rings $$T, R$$ where $$T$$ is the zero ring and $$R$$ is any arbitrary ring ... so $$T = \{ 0 \}$$ where $$0 = 1$$ ...
Then to me it seems that the "obvious" map is $$f( 0_T) = 0_R$$ ... ... which seems to me to be a ring homomorphism ...
... BUT ... this must be wrong ... but why ... ?
Can someone please clarify the above for me ...
Some help will be very much appreciated ...
Peter