In my previous sessional exams, I was asked to prove these two:
1) Find a ring which doesn't have a maximal Ideal.
2) If a ring has unity, then it has a maximal Ideal.
About the first one we can think about the trivial ring where there are only two additive subgroups (one the group itself and the other containing the identity element only....). Is there some other case for the first one...
About the second one I don't know how to show it...
Kindly help...
1) The simplest example is the zero ring. It has no maximal ideal (recall that maximal ideals are required to be proper ideals). For a more interesting example, consider the ring $(\mathbb{Q},+,0)$, where $+$ is the usual addition and $0$ is the zero multiplication. It is known that $(\mathbb{Q},+)$ has no maximal subgroups, which implies that $(\mathbb{Q},+,0)$ has no maximal ideals.
2) is not correct. As I've said, the zero ring (and this is unital with $0=1$) has no maximal ideal. But if $R$ is a non-zero unital ring, then $R$ has a maximal ideal. In fact, one may apply Zorn's Lemma to the partial order of proper ideals of $R$. The crucial observation is that for any chain $\mathcal{K}$ of proper ideals, their union $\bigcup \mathcal{K}$ is again a proper ideal. And this uses the existence of a unit and that an ideal is proper iff it does not contain the unit.