- #1
mahler1
- 217
- 0
Homework Statement
Find the zero divisors and the units of the quotient ring ##\mathbb Z[X]/<X^3>##
The attempt at a solution
If ##a \in \mathbb Z[X]/<X^3>## is a zero divisor, then there is ##b \neq 0_I## such that ##ab=0_I##. I think that the elements ##a=X+<X^3>## and ##b=X^2+<X^3>## are zero divisors because we have
##ab=XX^2+<X^3>=X^3+<X^3>=<X^3>##.
I couldn't think of any other divisors so I suspect these two are the only ones. Am I correct? If that is the case, how could I show these are the only zero divisors?
As for the units I don't know what to do. Any suggestions would be appreciated.
Find the zero divisors and the units of the quotient ring ##\mathbb Z[X]/<X^3>##
The attempt at a solution
If ##a \in \mathbb Z[X]/<X^3>## is a zero divisor, then there is ##b \neq 0_I## such that ##ab=0_I##. I think that the elements ##a=X+<X^3>## and ##b=X^2+<X^3>## are zero divisors because we have
##ab=XX^2+<X^3>=X^3+<X^3>=<X^3>##.
I couldn't think of any other divisors so I suspect these two are the only ones. Am I correct? If that is the case, how could I show these are the only zero divisors?
As for the units I don't know what to do. Any suggestions would be appreciated.