Prove that R is a ring with unity. 
Let $\langle R, +, \centerdot \rangle$ be a ring with identity element. Define the following operations $\oplus , \odot$ on $R$: 
  $a\oplus b = a+b+1$, $a\odot b = ab + a + b$
  Prove that $\langle R,\oplus,\odot\rangle$ is a ring with identity element. 

This feels like it should be pretty simple... I just don't know how to start.
 A: We could check one by one whether the properties that define a ring hold, starting with associativity of addition and multiplication. Each verification is mechanical, but there are quite a few to do. Painful!
Or  else let our new structure be $R^\ast$. Define a mapping $\varphi$ from $R$ to $R^\ast$ by $\varphi(x)=x-1$. Show that $\varphi$ is an isomorphism. 
A: The identity element for $\odot$ is 0. The identity element for $\oplus$ is -1. This is pretty easy to check. Once you have this, it's easy to check that the $\oplus$-inverse of $b$ is $-b-2$. The fact that it's a ring then follows from a tedious but straightforward verification that I'll leave to you.
Edit: How did I find the identity elements? Well in an arbitrary ring $R$, you can't say much about its elements, but in this case you're asked to find some specific elements of $R$, without relying on any properties of $R$. There are very few special elements of $R$, but amongst them are 0 and 1. Some checking shows that 0 is the identity for $\odot$, so you're naturally led to think, maybe 1 is the identity for $\oplus$. This turns out not to work, but the computation easily leads you to see that $-1$ is the identity for $\oplus$.
Another approach is this. You're asked to find an $x$ such that
$x + b + 1 = b$ for all $b$. Equivalently, $x + 1 = 0$ for all $b$, ie $x = -1$. Thus $x = -1$ is the identity for $\oplus$.
For $\odot$, you want to find $y$ such that $yb + y + b = b$ for all $b$. Equivalently, $yb + y = 0$ for all $b$, so $y(b+1) = 0$ for all $b$, so $y = 0$ is the identity for $\odot$.
You might think the second approach is more analytical, and hence more mathematical, and hence "better". But the first also has value. In particular, it's often useful to observe the limits of the information you are given. If you are given a lot of information in a problem, then you will probably need to use most of it. If you are given very little information, then there are only a few things that you can try!
