27 questions linked to/from Zero divisor in $R[x]$
5k views

### Zero divisors in polynomial rings [duplicate]

The following is an exercise in Hungerford (Ch. III, ex. 5.6). Let $R$ be a commutative ring with identity. If $f=a_nx^n+\dots+a_0$ is a zero divisor in $R[x]$, then there exists a nonzero $b$ in ...
81 views

### Zero divisor polynomial [duplicate]

Let $f(x)\in R[x]$ be a zero divisor. How to prove that there is an element $0\neq a\in R$ such that $af(x) = 0$? If $R$ has no nilpotent elements, it is easy. What about the general case? Can ...
45 views

### Let $A$ be a commutative ring and $f\in A[X]$ with $f \not\equiv 0$. Show that if $f|0$ in $A[X]$ then $\exists a \in A$ st $a \neq 0$ and $af = 0$ [duplicate]

For example, let $f= \sum_{i=0}^n a_ix^i$. If every $a_i\not\in U(A)$, then $\exists b_i\in A$ with $b_i \neq 0$ for every $i$ such that $a_ib_i=0$, so I can take $a = \prod b_i$ reaches what I need. ...
44 views

### Zero divisor for polynomial ring [duplicate]

I am having trouble with how to begin with this problem from Abstract Algebra by Dummit and Foote (2nd ed): Let $R$ be a commutative ring with 1. Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be ...
33 views

### Zero divisors in polynomial in a commutative ring [duplicate]

There's an exercise in Herstein marked as very hard Let $R$ be a commutative ring. If $q(x) \in R[x]$ be a zero divisor in $R[x]$, then if $\displaystyle q(x) = \sum_{0 \leq i \leq k} a_i x^i$, ...
20 views

157 views

### Prove that: $R[x]$ has a zero divisor $\Rightarrow$ $R$ has a zero divisor

The problem I've been trying to solve the above problem. There seems to already be some work regarding zero divisors in polynomial rings over here, but I'm not sure it is applicable to me, because it ...
801 views

### If $p$ is a prime ideal then $p[X]$ is a prime ideal

If $Z$ is a ring and $p$ is a prime ideal of $Z$ then $p[X]$ is a prime ideal of $Z[X]$. Is it true or false? I believe that it is true and I try to prove it like that: Take $f(x)\in p[X]$ and ...
### $R$ has nonzero nilpotent elements…
Let $R$ be a commutative ring with 1. a) Suppose $R$ has no nonzero nilpotent elements (that is, $a^n=0$ implies $a=0$). If $f(X)=a_0+a_1X+\cdots+a_nX^n$ in $R\left[X\right]$ is a zero-divisor, ...