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Describe the set of all endomorphisms of the additive group $\mathbb{Z}_{p}^{n}$ where $p$ is a prime. Under what operations is this set a ring?

It has been a while since I took Abstract Algebra and I am preparing for the prelims. I am not sure how to tackle this one. Any help/suggestion/hint will be much obliged.

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Is $\mathbb{Z}_p$ the $p$-adic integers or $\mathbb{Z}/p\mathbb{Z}$? –  Qiaochu Yuan Jun 25 '12 at 4:23
    
Yuan- That's an interesting question! I wrote down the problem as it was written. So, I am not sure! I have always used $\mathbb{Z}_p$ to mean integers modulo $p.$ –  Lyapunov Jun 25 '12 at 4:42

1 Answer 1

Hint 1. As an additive group, $\mathbb{Z}_p^n$ is a vector space over $\mathbb{F}_p$, so the additive group endomorphisms correspond to the vector space endomorphisms. How does one describe endomorphisms of a vector space?

The second question is a bit strange: any bijection between that set and a ring of that cardinality will give you a structure as a ring; I expect that they are looking for an obvious pair of operations that will make it into a ring in a natural way, and the first hint should tell you what one candidate might be.

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