
Re: Finite Rings
Posted:
Feb 4, 2013 8:02 AM


On Monday, February 4, 2013 10:26:16 AM UTC+2, William Elliot wrote: > On Sun, 3 Feb 2013, Arturo Magidin wrote: > > > On Sunday, February 3, 2013 10:46:14 PM UTC6, William Elliot wrote: > > > > On Sun, 3 Feb 2013, Arturo Magidin wrote: > > > > > On Sunday, February 3, 2013 9:21:19 PM UTC6, William Elliot wrote: > > > > > > > > > If R is a finite commutative ring without multiplicative identity > > > > > > > and if every element is a zero divisor, then does there exist > > > > > > > a nonzero element which annihilates all elements of the ring? > > > > > Which means that the trivial ring DOES NOT satisfy the hypothesis, and > > > therefore is not to be considered, period. The fact that a ring without > > > multiplicative identity must contain a nonzero element need not be a > > > premise, because "does not have a multiplicative identity" IMPLIES, > > > necessarily, the existence of a nonzero identity. > > > > Clever but no better. According to John Beachy in "Abstract Algebra," > > the multiplicative identity is distinct from the additive identity. > > > > So rather that toss bull about, what definitions are you using?
That Beachy example may not be that relevant here since many authors require
a ring NOT to be the zero ring, which is tantamount to requiring that the
neutral element and the multiplicative unit are different in the ring.
The fact here is that _IF_ we allow the zero ring, then it fulfills the
condition of having a multiplicative unit and, thus, cannot be taken as an
example.

