An example of a ring without identity that does not contain any maximal ideal.

Question

I'm trying to find an example of a ring without identity that does not contain any maximal ideal.

Help me some hints.

Answer 1

Take an abelian group without maximal subgroups, like $(\Bbb Q,+)$, and the zero multiplication, that is, $xy=0$ for any $x,y$.

Answer 2

Another example can be found in here http://sierra.nmsu.edu/morandi/notes/NoMaxIdeals.pdf