Examples?? Surjective (resp. Injective) ring endomorphisms which aren't isomorphisms

Question

Let f be an endomorphism on a ring R. (assumed unitary & commutative) We have the following results:

1. R noetherian & f surjective implies f an isomorphism
2. R artinian & f injective implies f an isomorphism

I'm interested in counterexamples to the above when R is not noetherian (resp. artinian).

Further, is there an example of a single ring R with 2 different endomorphisms, one injective and one surjective such that neither are isomorphisms.

My intuition is to perhaps look at infinitely generated algebras over a field/ pid/ "nice ring", however not sure of the details.