I'm trying to do a homework question and I must have something basic messed up in my head. I've looked up Trivial Ring and I've looked up Zero-divisor. A trivial Ring is a Ring with the single element 0. A Zero-divisor is a nonzero x such that ax = xa = 0.
When it says that a zero-divisor is a nonzero x such that ax = xa = 0, where are the x and a coming from? The x has to be nonzero and can come from anywhere, right? And a is from the ring in question? So if 0 can be a zero divisor for a non-trivial ring (a in the ring, x=0) then why can't it be a zero-divisor for the trivial ring? (a=the 0 in the ring, and x=0)?
Is it just saying that there is no nonzero element period, therefore you can't continue (leaving zero not a zero divisor)? That seems sloppy to me.
Can someone provide some clarification here?