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Abstract Algebra

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Moderate

Nilpotent Elements

ABSALG-GQVUNY

An element $x$ of ring $R$ is called nilpotent if $x^n = 0$ for some integer $n$.

What can be said about the nilpotent elements of a commutative ring $R$?

Select ALL that apply.

A

If $x$ and $y$ are nilpotent, then $x+y$ is nilpotent.

B

If $x$ and $y$ are nilpotent, then $xy$ is nilpotent.

C

If $x$ and $y$ are nilpotent, then $x-y$ is nilpotent.

D

The set of nilpotent elements form a subring of$R$.

E

The set of nilpotent elements for an ideal of $R$.