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.

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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