/sci/ - Science & Math

Suppose you have a graded ring [math]B[/math]. Is it required that the ring is non-empty in every degree?

For example, can I assume there exists an element [math]f\in B_1[/math]?

I'm asking because I'm reading this book that does not make any assumptions on the graded ring (other than it is an algebra over a field, but I don't think it's relevant here) and he's picking some element from [math]B_1[/math] for an argument. When reading Hartshorne, I think he always used to make some extra strong assumption like the irrelevant ideal [math]B_+[/math] is generated by [math]B_1[/math] so it wasn't be ambiguous, but he also used that property for other things (like finding an open cover with certain properties)