Articles of finitely generated

Are there any commutative rings in which no nonzero prime ideal is finitely generated?

Are there any commutative rings in which no nonzero prime ideal is finitely generated? I feel like the example (or proof of impossibility) ought to be obvious, but I’m not seeing it.

$M$ be a finitely generated module over commutative unital ring $R$ , $N,P$ submodules , $P\subseteq N \subseteq M$ and $M\cong P$ , is $M\cong N$?

Let $R$ be a commutative ring with unity , $M$ be a finitely generated module over $R$ , let $N,P$ be submodules of $M$ such that $P\subseteq N \subseteq M$ and $M\cong P$ , then is it true that $M\cong N$ ? If not true , then what happens if we also assume that $M$ […]

Example of subgroup of $\mathbb Q$ which is not finitely generated

I was looking for a proper subgroup of $\mathbb Q$ which is not finitely generated under the addition operation. We know every finitely generated subgroup of $\mathbb Q$ is cyclic. For a proper subgroup I am just thinking about the subgroup $H$ generated by $\{\frac{1}{p} : p \text{ prime }\}$ may work. It seems $1/4$ […]

Finitely generated graded modules over $K$

I need some help on this exercise from A Course in Ring Theory by Donald S. Passman Find all finitely generated graded $K[x]$-modules up to abstract isomorphism. Remember, $K[x]$ is a principal ideal domain. The result is supposedly similar to the well-known structure theorem in the non-graded case. So let $M$ be a finitely generated […]

For groups $A,B,C$, if $A\times B$ and $A\times C$ are isomorphic do we have $B$ isomorphic to $C$?

This question already has an answer here: $A\oplus C \cong B \oplus C$. Is $A \cong B$ when $C$ is finite, A and B infinite. 1 answer

How can I find an element $x\not\in\mathfrak mM_{\mathfrak m}$ for every maximal ideal $\mathfrak m$

Let $R$ be a commutative ring with finitely many maximal ideals $\mathfrak m_1,\ldots,\mathfrak m_n$. Let $M$ be a finitely generated module. Then there exists an element $x\in M$ such that $\frac{x}{1}\not\in\mathfrak m_iM_{{\mathfrak m}_i}$ for every $i=1,\dots,n$. I cannot prove that such an element there exists. I was trying to prove it by induction on $n$. […]

If $X$ is the set of all group elements of order $p$, and $X$ is finite, then $\langle X \rangle$ is finite

Suppose $G$ is any group and $X$ is the set of all elements of order $p$ in that group where $p$ is a prime. Prove that if $X$ is finite, then $\langle X\rangle$ is finite, where $\langle X\rangle$ is group generated by the elements of $X$. If possible, I’m interested in a solution that assumes […]

Any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic.

$\fbox{1}$ Prove that any finitely generated subgroup of $(\mathbb{Q},+)$ is cyclic. $\fbox{2}$ Prove that $\mathbb{Q}$ is not isomorphic to $\mathbb{Q}\times \mathbb{Q}$. Any hints would be appreciated.

If the tensor product of two modules is free of finite rank, then the modules are finitely generated and projective

If over a commutative ring $R$ we have that $M\otimes N=R^n$, $n\neq 0$, need we have that $M$ and $N$ are finitely generated projective? We have finite generation, because if $M\otimes N$ is generated by $\sum_i a_{ij} x_i^j\otimes y_i^j$, then we have $x_{i_1}^{j_1}\otimes y_{i_2}^{j_2}\otimes x^{j_3}_{i_3}$ generating $M^n$. Projecting onto $M$, we see that $x_i^j$ generate […]

the automorphism group of a finitely generated group

Let $G$ a finitely generated group, $\mathrm{Aut}(G)$ is its automorphism group, then it is necessary that $\mathrm{Aut}(G)$ is a finitely generated group? Thanks in advance.