Basic questions in Algebra Could someone confirm or check these out? Some of these are really simple, but I can't find a definite answer to these.

Let $R$ be a ring, $D$ an integral domain, $F$ a field, $G$ a group, and $N$ a normal subgroup of$G$


Do rings have to be closed under multiplication? My book only says multiplication needs to be associative.
Is this a proof that fields are integral domains? Let $x,y \in F - \{ 0 \}$. Assume $xy = 0 \iff xy + 1 = 1 \iff xy + xx^{-1} =1 \iff x(y + x^{-1}) = 1 \iff x^{-1} = y + x^{-1} \iff y = 0$, a contradiction, so no divisors of $0$, meaning $F$ is a $D$
Can someone briefly explain to me why the (first) Isomoprhism  theorems want to base on the Kernel so much? Without going the details, my intepretation of the theorems tells me we are just trying to cut up the group $G$ into ways we can understand it better thruogh the quotient, but I don't get why the general groundwork can't be used for arbitrary normal groups, why the kernel? My book does all these proofs, but I am not getting why the kernel is so special, why can't the isomoprhism theorems be based on some other normal groups. This is basically a case of "why" even though I know the "how".
1) Yes, rings have to be closed under multiplication.

2) That is a proof, but you can go simpler. Assume $xy = 0$. If $x = 0$, we're done. Otherwise $x \neq 0$ and so $x^{-1}$ exists. Then $y = x^{-1}xy = (x^{-1})(0) = 0$.

3) The First Isomorphism Theorem essentially states that given a homomorphism $\psi \colon G \to H$, then $G/\operatorname{ker}\psi \equiv \operatorname{im}\psi \leq H$. Basically, we use the kernel because it contains information about how $G$ is embedded into $H$. If you want the result with any other normal subgroup $N$, you'll have to find a homomorphism that has $N$ as the kernel. And there is, but it is trivial. Simply take $H = G/N$ and define $\psi(g) = gN$.
The Quotient Set G/H is made into a group by the multiplication

aHbH = abH

and

aHbH = abb^-1HbH = ab(b^-1Hb)H = abH

only if 

(b^-1Hb)  is a subgroup of H, that is,  only if H is normal., 

The kernel of any groupp hom is a normal subgroup of the domain; and any normal subgroup is the kernel of some group hom.