Linked Questions

5
votes
2answers
5k views

Finding invertible polynomials in polynomial ring $\mathbb{Z}_{n}[x]$ [duplicate]

Is there a method to find the units in $\mathbb{Z}_{n}[x]$? For instance, take $\mathbb{Z}_{4}$. How do we find all invertible polynomials in $\mathbb{Z}_{4}[x]$? Clearly $2x+1$ is one. What about ...
3
votes
3answers
7k views

What are the units of Z[x]? [duplicate]

Where $\mathbb{Z}[x]$ is the ring of polynomials in $x$ with integer coefficients. The book I am studying says the unity of this ring is $f(x) = 1$ so then if some $p \in \mathbb{Z}[x]$ is a unit, ...
5
votes
1answer
3k views

Units in a polynomial ring [duplicate]

I'm trying to determine $U(\mathbb{R}[x])$, where $U(R)$ denotes the unit group of a ring $R$. I think the answer is all non-zero constant polynomials, but I'm having trouble showing that these are ...
1
vote
2answers
2k views

Nilradical and Jacobson's radical. [duplicate]

Let A be a commutative ring with 1. 1) Prove that a sum of a nilpotent element and an invertible element is invertible. 2) Prove that if $f=a_0+a_1x+\dots+a_nx^n \in A[x]$ a) $\exists f^{-1}\in A[...
4
votes
1answer
284 views

The units of $\mathbb Z_4[x]$ [duplicate]

The units of $\mathbb Z_4[x]$ should be $$\{f(x)=a_0+a_1x+a_2x^2+\dots+a_nx^n\mid a_0\in\{1,3\},a_i\in\{0,2\}, 1\le i\le n\}.$$ But I don't know how to prove. Any suggestion?
0
votes
2answers
721 views

prove that $a_0$ is a unit and that $a_1 , a_2 , .. a_{n}$ are nilpotents in $R$ . [duplicate]

let R be a commutative ring . let $p(x) = a_n x^n + a_{n-1} x^{n-1} +...+a_1 x +a_0 $ $\in$ $R[x]$ prove that , $p(x)$ is a unit in R[x] iff a_0 is a unit and $a_1 , a_2 ,... , a_n $ are nilpotents ...
0
votes
0answers
112 views

$f$ is a unit in $R$ if and only if $a_0$ is a unit and $a_n, …, a_1$ are nilpotent [duplicate]

Prove that $f = a_nx^n + \cdots + a_1x+a_0$ is a unit in $R[x]$ if and only if $a_0$ is a unit in $R$ and $a_1, ..., a_n$ are nilpotent. My argument is as follows. Suppose that $f$ is a unit in $R[...
-2
votes
1answer
39 views

units of polynomial rings [duplicate]

When does a polynomial in the ring of polynomial have an inverse? I thought only constant polynomials were units. if there are other units, under what rings can we guarantee the existence of inverse ...
0
votes
0answers
36 views

Which are the units of $R[x]$? [duplicate]

In general be $R$ a ring. So what are $u(R[x])$? where $R[x]$ denotes the rings of polynomials with coefficients in $R$. For example $u(\mathbb{Z}[x]) = \{-1,1\}$
117
votes
14answers
63k views

Why is 1 not a prime number?

Why is $1$ not considered a prime number? Or, why is the definition of prime numbers given for integers greater than $1$?
8
votes
2answers
2k views

$x$ not nilpotent implies that there is a prime ideal not containing $x$.

Let $\mathscr{N}(R)$ denote the set of all nilpotent elements in a ring $R$. I have actually done an exercise which states that if $x \in \mathscr{N}(R)$, then $x$ is contained in every prime ideal ...
13
votes
1answer
5k views

Units of polynomial rings over a field

If $R$ is a field, what are the units of $R[X]$? My attempt: Let $f,g \in R[X]$ and $f(X)g(X)=1$. Then the only solution for the equation is both $f,g \in {R}$. So $U(R[X])=R$, exclude zero elements ...
2
votes
2answers
3k views

Under what conditions is a zero divisor element $a$ in commutative ring $R$ nilpotent?

Suppose that $R$ is a commutative ring with identity $1$ Let $a\in R$ with $ab=0$ for some $b\ne0$. Under what conditions $a$ must be also nilpotent?
6
votes
1answer
2k views

Find a polynomial of degree > 0 in $\mathbb Z_4[X]$ that is a unit.

Q: Find a polynomial of degree > 0 in $\mathbb Z_4[X]$ that is a unit. I know (2x+1) is a unit. Is there any other units in $Z_4[X]$ if there are infinite units, could you generalize it in a certain ...
4
votes
1answer
1k views

Proving a basic property of polynomial rings

I am learning ring theory in the Dummit & Foote's Abstract Algebra, and I am doing all the exercises to get as much experience as possible... but some of them just get me stuck for hours! Like ...

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