Abstract Algebra – Questions Related to Z[?-5] and UFD

could any one give me examples/proofs/counter examples against or for of the followings?

1.Homomorphic image of a UFD is again a UFD

2.The element $2\in\mathbb{Z}[\sqrt{-5}]$ is irreducible

3.Units of the ring $\mathbb{Z}[\sqrt{-5}]$ are units of $\mathbb{Z}$

4.$2$ is a prime element in $\mathbb{Z}[\sqrt{-5}]$

for 1, I know that $\mathbb{Z}$ is a UFD, to show the statement is false from $\mathbb{Z}$ to where I should construct a homomorphism?

for 2, In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.

3 is true.

4 is false