Let $R$ be a commutative ring with identity, and $I\neq 0$ an ideal of $R$, I’m thinking how to calculate $\text{End}_R(I)$. I have proved that when $R$ is a integral domain, $\text{End}_R(I)=\{r\in\text{Frac}(R)\mid rI\subset I\}$. So I’m curious about if $\text{End}_R(I)=R$ always holds when $R$ is also integrally closed. Can anyone help me with this? Thanks […]

Let $D$ be a UFD, let $F$ be the field of fractions of $D$, let $a \in D$ be such that $x^2 \ne a, \forall x \in D$. Then is it true that $x^2\ne a ,\forall x \in F$ ? (This problem is motivated from the fact that if square root of an integer is […]

Let $K$ be a field and $R_1,\dots,R_n$ DVRs of $K$ with $m_i$ the maximal ideal of $R_i$ and $R_i \not\subseteq R_j$ for $j\neq i$ . Define $A=\bigcap_{i=1}^n R_i$. Then $A$ is semilocal with maximal ideals $p_i=m_i\cap A$. Also, $A_{p_i}=R_i$. I know there is a $x_i \in p_i$ such that $x_i \not\in p_j$ for $i \neq […]

I tried to answer this question two days ago. Unfortunately, I said ring, rather than domain, which is what I wanted. So I try again. Let $R$ be a Noetherian commutative domain and let $r\in R$. Is it possible that $r$ have arbitrarily long factorizations? That is, is it possible that for all $n>0$, there […]

Let $R$ be an integrally closed domain and $S$ be an integral domain that contains $R$. Assume that $a\in S$ is integral over $R$. Prove that $I=\left\{ f\left(x\right)\in R\left[x\right]\mid f\left(a\right)=0\right\} $ is a principal ideal of $R[x].$ I only know that $R[x]$ is also an integrally closed integral domain and $a$ is a root of […]

I need some help to solve the second part of this problem. Also I will appreciate corrections about my solution to the first part. The problem is the following. Let $\sigma$ be an automorphism of the integral domain $R$. Show that $\sigma$ extends in an unique way to the integral closure of $R$ in it’s […]

Does there exist a Noetherian domain which is not a field , whose field of fractions is ( isomorphic with ) $\mathbb C$ ?

Consider any topological space $X$ and $\mathbb{R}$ be with usual topology. The set of all continuous functions from $X$ to $\mathbb{R}$, denoted by $C(X,\mathbb{R})$, is a commutative ring with unity under pointwise addition and pointwise multiplication. If we consider $X=[0,1]$ with usual topology, then $X$ is connected and $C(X,\mathbb{R})$ is not an integral domain. If […]

I need to show that being an integral domain is a local property. That is, a commutative ring $A$ has no zero divisors iff $A_{\mathfrak p}$ has no zero divisors for every prime ideal $\mathfrak p$. One way is obvious: if there are zero divisors in a localization, then there are zero divisors in $A$. […]

Let $R$ be a commutative ring. (i) Prove that $R$ has ACCP if and only if every non-empty collection of principal ideals of $R$ has a maximal element. (ii) Prove further that if $R$ is an integral domain and has ACCP, then $R[X]$ has ACCP. Attempt. (i) ($\Rightarrow$) Suppose that there exists a non-empty collection […]

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