Linked Questions

Possible Duplicate: Units and Nilpotents Given $A^{2012}=0$ prove that $A+I$ is invertible and find an expression for $(A+I)^{-1}$ in terms of $A$. ($I$ is the identity matrix).

Prove $1+x$ is a unit in $R: \text{commutative ring}$ where $x$ is nilpotent do I need to make use of a Taylor series expansion for this? $(1+x)(1+x)^{-1} = 1 \implies (1+x)^{-1} = \displaystyle\...

So my friend and I are working on this and here is what we have so far. We want to show that $\exists \, B$ s.t. $(I+A)B = I$. We considered the fact that $I - A^k = I$ for some positive $k$. Now, if ...

Possible Duplicate: Units and Nilpotents If $a$ is a unit and $x$ is nilpotent, I'm trying to show that $a+x$ is a unit. Pf.: If $a$ is a unit, there exists a non-zero invertible element $a^{-1}$ ...

Suppose that $a$ and $b$ belong to a commutative ring $R$. If $a$ is a unit of $R$ and $b^{2}=0$ . Show that $a+b$ is a unit of $R$ I try it.. consider $(a+b)(b)=ab+b^{2}=ab=1$ Since a is unit

If $A$ is a nilpotent matrix, then how to show that the matrix $I+A$ is invertible.

I have $R$ a commutative ring with unity and $a\in R$ such that $a^3=0$. How do I show that $1-a$ is a unit? If $a=0$, we are done, so I assume that $a\neq 0$, then can I say that $a$ and $a^2$ are ...

I can't tell where I went wrong, am still a beginner. Let N be a nilpotent in a commutive ring , let X be any element. I'll will be showing X + N is a unit . Assume Y(X+N) =1 Then YX(N^n-1) = ...

$T^{k+1} = 0$, where $k$ is a positive integer. Prove that $I + T + T^2 + \dots+ T^k = (I - T)^{-1}.$ My proof: Let $T$ be a $m \times n$ square matrix with the same number of $m$ rows and $n$ ...

Let $(A, +, \cdot)$ be a ring with $1$. An element $a\in A$ is nilpotent if there exists $n\in \mathbb{N}$ so that $a^n=0$. Show that if $a$ is nilpotent then $1+a$ and $1-a$ are invertible.

$Problem$ Suppose that a and b belong to a commutative ring $R$ with unity. If $a$ is the unit of $R$ and $b^2 = 0$; Show that $a+b$ is a unit of $R$. $ Attempt$ $(a+b)(a-b) a^{-2}= (a^2-b^2)a^{-2}=...

Find the multiplicative inverse of $1+2x$ in $\mathbb{Z_8}[x]$ My work: I know that since $1$ is a unit in $\mathbb{Z_8}[x]$ and $2$ is nilpotent of index $3$, $1 + 2x$ has a multiplicative inverse. ...

A non-zero matrix $A$ is said to be nilpotent for some positive integer $k\geq2$. If $A$ is nilpotent then is $I+A$ invertible?? Where $I$ is the identity matrix.

Let $R$ be a ring (not necessarily commutative). Let $u\in R^\times$, (so $u$ is a unit), and let $a\in R$ be a nilpotent element. Show that if $ua=au$ then $u+a\in R^\times$. I have that if $a$ is ...

Let $R$ be a commutative ring with $1$. Let $a\in R^\times$ and $b\in\text{Nil}(R)$. Show that $a+b\in R^\times$. Attempts: Suppose $b^n=0$. I've tried to figure out a "simple" case as $n=2$; I've ...