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 $\displaystyle z^+$ say whether or not it is associative, commutative

i) $\displaystyle a*b=a+b^2$

This is neither commutative nor associative

ii)$\displaystyle a*b=a+1 $

Is again neither commutative nor associative

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

I think $\displaystyle a*b =$ $\displaystyle a^2+b^2$ or $\displaystyle a*b=ab+1$

c) On the set of integers Z define a multiplication $\displaystyle *$ by $\displaystyle a*b=1+b$ for any $\displaystyle a,b \in Z$ . Show that <Z,+,*> is not a ring, where + is the usual addition.

I think this fails because $\displaystyle (a*b)*c=1+c$, but $\displaystyle a*(b*c)= 2+c$ 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 $\displaystyle a+b=b+a=0$.

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

Then $\displaystyle (a+b)*c=0$, but $\displaystyle (b+c)*a=a*b+a*c$ 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!