TITLE: Pathologies in "rng"
QUESTION [22 upvotes]: There is no general consensus regarding whether a ring should have a unity element or not. Many authors work with unital rings , and other does not essentially require unity. 
If we do not assume unity to be a necessary part of ring, lets call that structure , a "rng" (which may or may not have unity i.e. $1$), then many pathologies do occur. 
I am hoping to get such pathologies (with at least one example) listed together on a page which I can keep as a record for future references. 
I will post my list as an answer here, to get it started, and hope others will contribute to it, and make this post valuable.

REPLY [2 votes]: It seems like no one has mentioned the pathology that first comes to my mind for rings without identity:

A ring without identity may not have maximal (left/right/two-sided) ideals.

The fact that a ring with identity must have maximal ideals is a sort of very weak finiteness condition. Of course, it is not a spectacular finiteness condition since there are rings without identity that have maximal ideals anyway (even finite rings without identity) but still it is a special condition.
Example- $(\Bbb{Q},+,\circ)$ where $\circ$ is trivial multiplication has no maximal ideals as ideals in this rng are same as subgroups of $(\Bbb{Q},+)$ and we can prove that $(\Bbb{Q},+)$ has no maximal subgroups.