Question

Give an example of infinite ring of characteristic 2

Answer 1

Consider the infinite ring $(\mathbb Z_2,+,\cdot)^{\omega}=\{(a_0,a_1,a_2,...,a_n,...)|\ a_i\in\Z_2\},$ where $\Z_2=\{0,1\}.$

Taking into account that

$2\cdot(a_0,a_1,a_2,...,a_n,...)=(2a_0,2a_1,2a_2,...,2a_n,...)=(0,0,0,...,0,...)$

for any $(a_0,a_1,a_2,...,a_n,...)\in(\mathbb Z_2)^{\omega}$,

we conclude that this ring is of characteristic 2.