localisation of ring

[mona123]

Hi. Can someone please help me to prove this equivalence?

Let A be a commutative ring and "a" is in A. We note A_a the localisation of A by S = {aⁿ, n dans N}.
Knowing that A[x]/(ax-1) is isomorphic to A_a, show that A_a #0 if and only if a is not nilpotent

Thanks in advance.

 

[stapel]

Let A be a commutative ring and "a" is in A. We note A_a the localisation of A by S = {aⁿ, n dans N}.
Knowing that A[x]/(ax-1) is isomorphic to A_a, show that A_a #0 if and only if a is not nilpotent

What does "dans" mean? Are you using the "bash" character ("#") to indicate "is not equal to"? Are you using the "zero" following the bash to indicate "the empty set"?

When you reply, please include a clear listing of your thoughts and efforts so far. Thank you!