ABSTRACT
Consider Example 6.7: We discovered that it was relatively easy to show that Z[x] is a ring, because it is a subset of Q[x], which we had already shown is a ring. Because the operations in Z[x] are the same as in Q[x], we didn’t have to check again the associative laws, the distributive laws, or that addition was commutative. Because the addition and multiplication of Q[x] have these properties, the addition and multiplication of Z[x] inherit them automatically. What did need to be checked was that addition and multiplication were closed in Z[x], that the additive identity of Q[x] was also in Z[x], and that the additive inverses of elements of Z[x] were also in Z[x]. Similarly, in Example 6.8 you showed that 2Z is a ring, taking advantage of the fact that 2Z ⊆ Z.
