Let R be a ring and N the set of nilpotent elements,

i.e.,N = {x ∈ R | $$x^n$$​ = 0 for some n ∈ N}.

Which of the following is true?

Correct option is D

Let R be the ring, andN = {x ∈ R | $$x^n$$​ = 0 for some n ∈ N}.

To check whether N is an ideal, the following conditions must hold:

1. For all a, b ∈ N, ab ∈ N.

2. For all r ∈ R and a ∈ N, ra ∈ N and ar ∈ N.

Now, let a, b ∈ N and r ∈ R.

Then: $$a^n=0,b^m=0$$​ for some n, m ∈ N.

$$(ar)^n = (a^n)r^n = 0 \cdot r^n = 0 \implies ar \in N.$$​

 Similarly, $$(ra)^n = r^n a^n = r^n \cdot 0 = 0 \implies ra \in N.$$​

 Next, consider ab:

(ab)$$^n$$​ = (ab)(ab)(ab)· · ·(ab) (n times).This need not be zero in general unless R is commutative.

If R is commutative: $$(ab)^n = a^n b^n = 0 \implies ab \in N.$$​​

Thus, N can be an ideal of R provided R is commutative.