Prove that $(\mathbb Z_{10},∔,⨰)$ is a ring where the operations $∔$ and $⨰$, $\mathbb Z_{10}×\mathbb Z_{10}→\mathbb Z_{10}$ Starting ring and field theory and need a hand understanding how to prove a ring under its axioms. I have to prove $(\mathbb Z_{10},∔,⨰)$ is a ring where
$$a∔b=a+b+7,\quad a⨰b=ab+7(a+b)+2.$$
So far I have that it must be an abelian group, so associative, commutative. have an additive identity and an additive inverse (A1), i.e.
$$(a∔b)∔c=a∔(b∔c).$$
LHS: $$(a∔b)∔c=(a+b+7)∔c=a+b+c+7.$$
RHS: $$a∔(b∔c)=a∔(b+c+7)=a+b+c+7.$$ 
so it is associative.
Commutative: $$a∔b=b∔a.$$
RHS:
$$b∔a=b+a+7.$$
LHS:
$$a∔b=a+b+7.$$
So commutative.
I am not entirely sure how to show the identities as surely it is obvious that $a∔0=0$ and $a∔(-a)=0$?
Then I think axiom 2 and 3 are fine. I just do not understand how to show the basics here.
 A: 
I am not entirely sure how to show the identities as surely it is obvious that a∔0=0 and a∔(−a)=0?

Obvious as in without thinking we will just say so?  Tempting, perhaps, but not a very good habit. Or obvious if you think $0\dotplus 0 = 0+0+7=0$, so that $7=0$. Perhaps not so obvious.
To find the additive and multiplicative identities (if they exist!) you should just write out what that means, by which I mean
$a\dotplus z=a+z+7=a$ and $a\dot\times e=ae+7(a+e)+2=a$ and then take a look. There are only 10 elements in this case, after all.

There is a different way to prove this, one that's used multiple times throughout the site in similar questions. Did you encounter them when searching for your question, initially?
Let $\phi:\mathbb Z_{10}\to \mathbb Z_{10}$ be given by $x\mapsto x+3$. Clearly this map is a bijection of sets.
Moreover, you can check that 
$\phi(a+b)=\phi(a)\dotplus\phi(b)$ and $\phi(ab)=\phi(a)\dot\times\phi(b)$
From here it is trivial to verify that because $+,\times$ constitute ring operations on $\mathbb Z_{10}$, then $\dotplus, \dot\times$ constitute ring operations on $\phi(\mathbb Z_{10})=\mathbb Z_{10}$, and in fact $\phi$ is an isomorphism of rings viewed this way.
This is called "transport of structure."  The idea is that any bijection of a ring with a set can be used to "force" a ring structure upon the set simply by multiplying things the way they behave in the ring whose structure we are transporting.
