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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:

In 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:

In 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