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# B. Associativity and Distributivity of Operations in

Let satisfy . Let . Let and be the binary operations on defined by

Thus for all ,
 (B.1)

 (B.2)

We will show that and are associative by using the usual properties of addition and multiplication on .

B.3   Lemma. Let , . If , then and .

Proof:

Case 1. Suppose . Then by our assumptions,

and

So

Since , it follows that and since is an integer , so . Then , so .
Case 2. If , use Case 1 with and interchanged.

B.4   Theorem. is associative on .

Proof: Let . Then

 (B.5)

 (B.6)

 (B.7)

 (B.8)

By adding to both sides of (B.5), we get
 (B.9)

and by adding to both sides of (B.6), we get
 (B.10)

Replace in (B.9) by its value from (B.7) to get
 (B.11)

and replace in (B.10) by its value from (B.8) to get
 (B.12)

By (B.11) and (B.12) and the associative law in ,

the associativity of follows from lemma (B.3).

B.13   Theorem. is associative on .

Proof: The proof is nearly identical with the proof that is associative.

B.14   Theorem. The distributive law holds in ; i.e., for all ,

Proof: We have

 (B.15)

 (B.16)

 (B.17)

 (B.18)

Multiply both sides of (B.15) by to get
 (B.19)

Replace in (B.19) by its value from (B.16) to get
 (B.20)

Now add equations (B.17) and (B.18) to get
 (B.21)

We know that for some ,

and if we substitute this into (B.21), we obtain
 (B.22)

From (B.20) and (B.22) and the distributive law in , we conclude

The distributive law follows from lemma B.3.

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