commutative ring and unity elements proof So this is a review problem in our book I came across and i really want to understand it but I am just not having any luck, I did some research and found a guide on solving it but that's not really helping either. We didn't talk about unity elements in class and there aren't any examples in our book just this problem. Would anyone be willing to take the time to fully explain this to me and show me how to prove it?
Question:
You may assume that 
E=even integers is a Commutative Ring. 
Prove that E does NOT have a unity element 
Information I found online: It suffices to show that there is at least a single element, n, of E , for which no element of E acts as a unity element for that specific n. 
So specifically, consider the case n = 2 and let m = any element of 
E. Then m MUST have exactly 1 of 3 possible properties: 
so let CASE 1 cover the 1st possible property of m and show that m cannot be a “unity” for 2 
and let CASE 2 cover the 2nd possible property of m and show that m cannot be a “unity” for 2 
and let CASE 3 cover the 3rd possible property of m and show that m cannot be a “unity” for 2 
Conclude that if there is no unity element ‘e’ for 2 such that 2*e=e*2=2
then, of course, there is no unity element for all of 
E … 
E does NOT have a unity element
SO
How can I use this information to formulate a proof and can anyone explain what exactly unity elements are in simpler language than my book and wikipedia use?
What I've worked out
A unity element e must satisfy er=r for every r∈R. It suffices to show that er=r is impossible when r=2. Since any candidate e is an even integer, you can write it as e=2k for some integer k.
Am I going the right direction? Where can I go from here?
 A: An element $1_R$ in a ring $R$ is called the unity element if for every $r \in R$, we have that $$1_R r = r = r 1_R$$
So the task is to prove that there is no such element in the ring $E$ of even integers. Suppose there was; then you'd have $$1_E r = r$$
for every $r \in E$. More particularly, we'd even have $1_E 2 = 2$. Now since $1_E$ can be written as $2k$ for some integer $k$, we have
$$(2k)(2) = 2$$
as a statement about integers. Thinking about this in the ring of integers, this implies that $2l = 1$, which is obviously false for any integer $l$, so we've reached a contradiction.
A: Essentially we're looking at a set of abstract concepts that we call commutative rings without unity. If you're thinking philosophically about why we are looking at these cases the answer is that if it exists it's worth exploring. The reasons it's worth exploring is because just about everything in mathematics follows or has a set of properties. The properties we are taught in grade school and even in some undergraduate courses are very few and apply to a wide range of subjects that are taught exclusively in those courses. When looking at some of the higher level mathematics there is less structure unless specifically stated. For example, if you say "Plot the function x on a graph" a college algebra student will draw a diagonal line on a Cartesian Plane that extends indefinitely in both directions whereas someone in higher level mathematics may ask for more information before making the attempt. There is an arrogance to thinking that some things should be left unexplored and most mathematicians will advise any aspiring mathematician to absolve themselves of that arrogance. 
Another simple answer is that if we look at commutative rings without unity and ask questions such as this one it forces the person being challenged to take the information he/she has learned and apply it in a different way. Very few challenges in everyday life will be of the same form. So take your lessons and apply them where you can. i.e. take what you know of commutative rings with unity and see what changes for commutative rings without unity. Because you never know when it'll come up! 
A: Well, the set of even integers $E$ forms a subring of the ring of integers ${\Bbb Z}$ if you don't insist that the unit element $1_{\Bbb Z}\in{\Bbb Z}$ is inherited for a moment.
The unit element $1_E$ in $E$ must fulfill $r\cdot 1_E = r = 1_E\cdot r$ for each $r\in E$. Now consider the cancellation property of ${\Bbb Z}$:
$$a\cdot b = a\cdot c\Rightarrow b=c\mbox{ whenever $a\ne 0$.}$$
This property is inherited by the subring $E$. So for each $r\in E$,
$$r\cdot 1_E = r = r\cdot 1_{\Bbb Z}\Rightarrow 1_E = 1_{\Bbb Z}.$$
Thus $E$ cannot have a unit element.
