Q:A ring R has maximum ideals
a)if R is infinite
b)if R is finite
c)if R is finite atleast 2 elements
d)only if R is finite
Is only one allowed to be true? I havent thought about the others, and I cannot be positive that all infinite rings posses maximum ideals, but certainly the following does.
Consider a polynomial ring with coefficients from a field $\displaystyle \mathbb{F}$. $\displaystyle \mathbb{F}[x]$ is an infinite ring a Principal Ideal Domain and any irreducible polynomial forms a maximal ideal since in PIDs irreducible implies prime which implies maximal.