algebra homomorphism

Sep 2019
Can anyone offer some help? Thank you for your time.

Let $\varphi :A\to B$ be an algebra homomorphism.

(a) The inclusion $\varphi \left( Z(A) \right)\subseteq Z(B)$ does not always hold. To see why, give an example of when it does hold and another example of when it does not hold.

(b) Come up with exactly one property of the map $\varphi $, where $\varphi $ could be any algebra homomorphism from $A$ to $B$ but the identity map, such that the inclusion in question 2(a) always holds and prove that this inclusion is then true.
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Sep 2016
So you have tried nothing and you are all out of ideas? Can you really not come up with a single attempt here?

How about write down any algebra homomorphism you want. Either (a) holds or it doesn't, so now you have half the work done. Then analyze whatever example you have and try to figure out why it's true or not.

I'm happy to provide more hints if you show that you have put some effort into the problem. I'm not willing to do your homework for you though.
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