Guided Example #3 - Algebraic Structures

We start by showing that S is closed with respect to addition.
For ${\displaystyle m+n\sqrt[3]{7}+p\sqrt[3]{3},x+y\sqrt[3]{7}+z\sqrt[3]{3\in S}}$,
${\displaystyle m+n\sqrt[3]{7}+p\sqrt[3]{3}+x+y\sqrt[3]{7}+z\sqrt[3]{3}=(m+x)+(n+y)\sqrt[3]{7}+(p+z)\sqrt[3]{3}\in S}$ .

Then ${\displaystyle -a\ge 0}$ and ${\displaystyle -a+a\ge 0+a}$ (Addition property of order)

Therefore, the set S is closed under addition.
Now we proceed to show that S is closed under multiplication.
${\displaystyle (m+n\sqrt[3]{7}+p\sqrt[3]{3})(x+y\sqrt[3]{7}+z\sqrt[3]{3})}$
${\displaystyle =(mx+3nz+3py)+(my+nm+3pz)\sqrt[3]{7}+(mz+ny+px)\sqrt[3]{3\in S}}$
So S is also closed under multiplication.

Next note that since S is a subset of the ring R the following properties hold.

Finally, ${\displaystyle 0+0\sqrt[3]{7}+0\sqrt[3]{3}=0\in S}$ , and for each ${\displaystyle a+b\sqrt[3]{7}+c\sqrt[3]{3}}$ there exists
${\displaystyle -m-n\sqrt[3]{7}-p\sqrt[3]{3}\in S}$. Hence S has all the required properties of the ring.