Integral Domain

May 2009
86
1
Hello I am just trying to work out some problems from herstein. Can you please tell me if this one is right.


If D is an integral domian and if na=0 for some a not 0 in D and for some integer n not 0 , prove that D is of finite characterstic.

so we have for some a not 0, na = 0
if b not 0 is another element in D then nab = 0b (ie) (na)b = 0

na is an element of D, an integral domain which has zero divisors, so(na)b=0 where na = 0 is justified.

However nab=0 => (nb)a=0
Here nb is an element of D and a is nonzero,
So nb = 0.

Hence the proof.

Am i right?
 
Last edited:
May 2009
1,176
412
Hello I am just trying to work out some problems from herstein. Can you please tell me if this one is right.


If D is an integral domian and if na=0 for some a not 0 in D and for some integer n not 0 , prove that D is of finite characterstic.

so we have for some a not 0, na = 0
if b not 0 is another element in D then nab = 0b (ie) (na)b = 0

na is an element of D, an integral domain which has zero divisors, so(na)b=0 where na = 0 is justified.

However nab=0 => (nb)a=0
Here nb is an element of D and a is nonzero,
So nb = 0.

Hence the proof.

Am i right?
That looks good to me.