Answer to Question #138971 in Abstract Algebra for Prabir khurana

Question

Give an example of infinite ring of characteristic 7.

Answer 1

Consider the ring $\Bbb Z_7[x]$ of polynomials in one variable 𝑥 with coefficients in ℤ$_7$ It is an infinite ring since $x^m \in \Bbb Z_7[x]:=\{a_1x^m+a_2x^{m-1}+...+a_n:a_1\ne 0,a_1,...,a_n\in \Bbb Z_7\& m\in \Bbb Z^+\}$ and note that for all positive integers 𝑚, and $x^{m_1} \ne x^{m_2}$ for $m_1\ne m_2$ ,Thus $Cardinality( \Bbb Z_7[x])=\infty$ But the charactetistic of $\Bbb Z_7[x]$ is 7 as 7 is prime and 1+1+1+1+1+1+1=7=0. in the ring $\Bbb Z_7$ .