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**r45** Hi there, I'm stuck on a couple of questions about integral rings:

1. State the difference(s) between an integral ring and an integral domain.

2. Suppose R is an integral ring, and x, y, z are 3 elements in R. Furthermore, suppose $\displaystyle x \neq 0$ and xy = xz. Show that y = z.

3. Again, suppose R is an integral ring, and x is a non-zero element in R. Is the map of x into ax an injective mapping of R into R? Why/why not?

For 1, is an integral domain simply an integral ring in which every element is commutative? Or is there something else that differentiates the two?

For 2, I am confused, at first glance it seems like an obvious result but I've been struggling to show it. The fact that R is an integral ring (as opposed to an integral domain) means that x, y, z may not have a multiplicative inverse. So I cannot just multiply xy = xz on the left by $\displaystyle x^{-1}$ to get y = z. I've tried using the associativity axiom of rings etc but I haven't been able to get the desired result, what am I missing?!

And for 3, I think the answer is yes (that's the intuitive answer, at least - although by now I've realised that by no means does that mean that it's the right answer!) - but why, can you show why it is an injective mapping?

Many thanks for any help!