Tagged Questions
0
votes
1answer
46 views
“Two ways” to reduce a module
Let $M$ be a module over a principal ideal domain $R$ and $\mathfrak{m}$ a maximal ideal of $R$ with residue field $R/\mathfrak{m}=k$ of characteristic $p$.
Under what circumstances are the modules
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2
votes
1answer
65 views
Reference on Operators on Modules over PID
The operators on finite dimensional vector spaces over any field can be seen in nice form w.r.t. some choice of basis such as "diagonal, triangular, Jordan form, rational form etc." But, for ...
2
votes
0answers
190 views
Exercise on modules over PID involving injective modules, Baer's criterion.
I'm interested if I solved this somewhat correctly, and would like to be set straight if it is wrong. This is an exercise from an introductory text.
Let $A$ be a module over a principal ideal ...
2
votes
1answer
109 views
How can we write this $\mathbb{Q}[x]$-module as a direct sum of cyclic $\mathbb{Q}[x]$-modules?
If $L$ is the submodule of $\mathbb{Q}[x]^{(3)}$ generated by $(2x-1,x,x),(x,x,x),(x+1,2x,x)$. How do we write $\mathbb{Q}[x]^{(3)}/L$ as a direct sum of cyclic modules?
3
votes
1answer
98 views
Do a matrix and its transpose have the same invariant factors over a PID?
I suspect this is true since it holds in the case over a field. But suppose $A\in M_{m\times n}(R)$ where $R$ is a PID. Does it still hold that $A$ and $A^{T}$ have the same invariant factors?
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2
votes
2answers
104 views
Sub-module over the Fraction field of a PID
R is a PID with field of fractions K. $M \subseteq K$ is a fin. generated R-Submodule. I am trying to show M is in fact generated by one element.
-1
votes
1answer
232 views
Isomorphism between set of all R-modules homomorphisms
Let $S$ be an $R$-module where $R$ is an integral domain and let $P = \langle p \rangle $ be a prime ideal.
Define:
$S_{P}=\{s \in S: p^{n} s=0 \ \textrm{for some natural }n\}$.
As usual, denote ...
10
votes
1answer
279 views
Finitely generated modules over PID
Let $A$, $B$, $C$, and $D$ be finitely generated modules over a PID $R$ such that $A\oplus $ $B$ $\cong$ $C\oplus $ $D$ and $A\oplus $ $D$ $\cong$ $C\oplus $ $B$ . Prove that $A$ $\cong$ $C$ and $B$ ...
