A commutative ring \(R\) with identity is called an integral domain if, for every \(a, b \in R\) such that \(ab = 0\text{,}\) either \(a = 0\) or \(b = 0\text{.}\) A division ring is a ring \(R\text{,...A commutative ring \(R\) with identity is called an integral domain if, for every \(a, b \in R\) such that \(ab = 0\text{,}\) either \(a = 0\) or \(b = 0\text{.}\) A division ring is a ring \(R\text{,}\) with an identity, in which every nonzero element in \(R\) is a unit; that is, for each \(a \in R\) with \(a \neq 0\text{,}\) there exists a unique element \(a^{-1}\) such that \(a^{-1} a = a a^{-1} = 1\text{.}\) A commutative division ring is called a field.
In the previous section, we observed that many familiar number systems are fields but that some are not. As we will see, these non-fields are often more structurally interesting, at least from the per...In the previous section, we observed that many familiar number systems are fields but that some are not. As we will see, these non-fields are often more structurally interesting, at least from the perspective of factorization; thus, in this section, we explore them in more detail. Before we proceed with that endeavor we will give a formal definition of polynomial so that we can include it in our work.
