Hi I've got some problems and just need a bit of reassurance that I'm doing them right!

a) For each of the following binary operations on say whether or not it is associative, commutative

i)

This is neither commutative nor associative

ii)

Is again neither commutative nor associative

b)Give an example of a binary operation on Z which is commutative but not associative

I think or

c) On the set of integers Z define a multiplication by for any . Show that <Z,+,*> is not a ring, where + is the usual addition.

I think this fails because , but so not associative under multiplication. However this is for 6 marks so feel like I must be missing something!

d) Prove the assertion using only axioms of a ring. Let R be a ring with a an element in R. Show there is exactly one element in which .

For this I assume two distinct elements b and c satisfy this i.e.

Then , but not zero so assumption fails. Not sure if this is precise enough??

e) Is it true that every subring in a commutative domain R with unity is an ideal in R? Prove your claim.

I think this is true but no idea how to prove it

Sorry I know this is quite a few questions just want a bit of verification that what I'm doing is right and a bit of help with the last question!