A Nil Ideal that is not Nilpotent

A Nil Ideal that is not Nilpotent

An ideal  I \leq R is said to be nil is each of elements is nilpotent.  I is said to be nilpotent if there exists some  k such that  I^k =0. That is, every product of  k elements of  I is zero.

Consider the ring  R = \mathbb{C}[x_1,x_2,x_3,...]/(x_1,x^2_2,x^3_3,...). Then, the ideal  I = (x_1,x_2,x_3,...) is nil in  R because each of its generators is nilpotent.

Moreover,  I is not nilpotent. To see this, assume by contradiction that it is. Then, there exists some k > 0 such that I^k=0. This implies that for all  i,  x_i^k=0 in  R . However,  x_{k+1}^k \neq 0 in  R . This is the desired contradiction.

 

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