Re: Divisors of zero in the quotient ring Z[X]/(X^2, 4)
- From: "Mariano Suárez-Alvarez" <mariano.suarezalvarez@xxxxxxxxx>
- Date: Thu, 3 Jan 2008 08:10:20 -0800 (PST)
On Jan 3, 12:47 pm, precarion <precar...@xxxxxxxx> wrote:
Hello! :)
Can anyone explain to me what are the divisors of zero in the quotient ring Z[X]/(X^2, 4)?
Chris
Let R = Z[X]/(X^2,4). It is quite evident that the classes
of X and and 2 are evidently nilpotent, so they are divisors
of zero. Moreover, the image I in R of the ideal (X, 2) is then
entirely composed of nilpotent elements. Since R/I is a field,
then I is precisely the set of nilpotent elements and, in this
case, it is clear that it contains all divisors of zero.
In general, if J is an ideal in a commutative ring R such that
the radical r(J) is prime, then r(J)/J is the set of divisors
of zero in R/J.
-- m
.
- References:
- Divisors of zero in the quotient ring Z[X]/(X^2, 4)
- From: precarion
- Divisors of zero in the quotient ring Z[X]/(X^2, 4)
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