## Is the Set of Nilpotent Element an Ideal?

## Problem 620

Is it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$?

If so, prove it. Otherwise give a counterexample.

Add to solve laterIs it true that a set of nilpotent elements in a ring $R$ is an ideal of $R$?

If so, prove it. Otherwise give a counterexample.

Add to solve laterLet $R$ be a commutative ring with $1$ such that every element $x$ in $R$ is idempotent, that is, $x^2=x$. (Such a ring is called a **Boolean ring**.)

**(a)** Prove that $x^n=x$ for any positive integer $n$.

**(b)** Prove that $R$ does not have a nonzero nilpotent element.

A square matrix $A$ is called **nilpotent** if some power of $A$ is the zero matrix.

Namely, $A$ is nilpotent if there exists a positive integer $k$ such that $A^k=O$, where $O$ is the zero matrix.

Suppose that $A$ is a nilpotent matrix and let $B$ be an invertible matrix of the same size as $A$.

Is the matrix $B-A$ invertible? If so prove it. Otherwise, give a counterexample.

Let $R$ be a ring with $1$.

Suppose that $a, b$ are elements in $R$ such that

\[ab=1 \text{ and } ba\neq 1.\]

**(a)** Prove that $1-ba$ is idempotent.

**(b)** Prove that $b^n(1-ba)$ is nilpotent for each positive integer $n$.

**(c)** Prove that the ring $R$ has infinitely many nilpotent elements.

Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.

Add to solve later Let $R$ be a commutative ring with $1 \neq 0$.

An element $a\in R$ is called **nilpotent** if $a^n=0$ for some positive integer $n$.

Then prove that if $a$ is a nilpotent element of $R$, then $1-ab$ is a unit for all $b \in R$.

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