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## Homework Statement

Taken from a Ring theory class:

Let ##A## be the ring of decimal number:$$ A= \{ n10^k|n \in Z, k \in Z \}$$

Show that ##x \in A## if and only if it's decimal expansion is finite. Here we assume that ##x \in R##.

## Homework Equations

I have no relevant equations really except all the basic definitions of rings which I don't think is useful here.

## The Attempt at a Solution

I see it intuitively why but I have a hard time proving it. My attempt is something like:

##\Rightarrow## :

Since ##x \in A##, we can write ##x = n * 10^k## for ##k,n \in Z##.Therefore the decimal expansion will have at most a term of ##k##th power. Therefore, the decimal will have at most ##k + 1## terms which means it is finite.

##\Leftarrow## :

Since the decimal expansion of x has a finite number of terms, we can multiply ##x## by a power of 10, ##(10^h)##, high enough that ##x 10^h \in N##.

Is that a sufficient enough proof? If feel like the right implication is close but not there for some reason.

Thanks