I keep reading questions like: "prove that this ring has a unit element under such axioms" or "prove that morphisms of rings preserve the unit element under those circumstances"
I find these questions weird because I would have thought that any moderately recent document (textbook, article, ...) incorporates those requirements on units in the definition of ring and ring morphism, implicitly or explicitly.
First question: Are there any counterexamples, for example authoritative recent ($\leq 25$ years) textbooks defining rings without requiring unit elements ? Or do I suffer from the diseases bourbakitis and algebrogeometritis?
Second question: Should we tell questioners that their terminology is not optimal (in a sociological , not a logical sense of course) and advise them to use Jacobson's happy terminology "rngs" in the absence of unit, or point out that their "rings" are actually ideals of genuine rings.
That said, answerers would of course be encouraged to answer the OP's question, after this little terminological admonition.
Of course, you might say: "Do as you like when you answer and don't bother us, Georges"
This is indeed excellent advice, but the point is that I tend not to answer these questions (I concentrate on other tags) and nevertheless would like users to encourage the usage I advocate if indeed they agree.
Well, my question has been answered : yes, some very active users here defend the definition in which rings may not have a unity .
So that we need no common policy on answers to related questions, and I suppose we will all go on using the definition adopted by the mathematical tribe we belong to: no harm in that!
Thanks to you all for your contributions .