Articles of modules

A question about primary decomposition and invariant factor decomposition

Find the primary decomposition and invariant factor decomposition of $\mathbb{Z}/180\mathbb{Z}$. For the invariant factor decomposition, we just find the prime factorization of $180$ and write $180=5\times3\times3\times2\times2$. So $\mathbb{Z}/180\mathbb{Z} = \mathbb{Z}/6\mathbb{Z} \oplus \mathbb{Z}/30\mathbb{Z}$, right? But what if we said, for example, $\mathbb{Z}/180\mathbb{Z} = \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/90\mathbb{Z}$ or $\mathbb{Z}/180\mathbb{Z}= \mathbb{Z}/3\mathbb{Z} \oplus \mathbb{Z}/60\mathbb{Z}$. Would those aslo count as […]

Support of $\operatorname{Hom}(R/I, M)$

Let $R$ be a Noetherian ring, $I$ be an ideal of $R$ and $M$ be an $R$-module. Is the following formula true? $\operatorname{Supp}\operatorname{Hom}(R/I, M)=\operatorname{Supp}(M) \cap V(I)$ If not, under what conditions the above formula is true? Thanks.

Non-Examples for Krull-Schmidt-Azumaya

I am looking for some rings $A$ and finitely generated $A$-modules $M$, where the conclusion of the Krull-Schmidt-Azumaya Theorem does not hold for $M$, i.e. where $M$ can be written as direct sums of indecomposable modules in multiple ways (the summands not being unique up to order and isomorphism). To be specific, I am looking […]

If $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules and if $M$ is a maximal ideal of $R$ then how can I show that image of $M{^m}$ is $M{^n}$?

If $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules and if $M$ is a maximal ideal of $R$ then how can I show that image of $M{^m}$ is $M{^n}$? Background: I was trying to prove that if $R{^m}$ is isomorphic to $R{^n}$ as $R$-modules then $m=n$. For this I need a step like $R{^m/M{^m}}$ isomorphic to […]

book suggestion on module theory

I want to start module theory from the very beginning. I have a little bit of idea about what module is but I haven’t really solved a handful of problems. Could someone please suggest me a good treatise on this subject? I badly need it.

How can one show that ${\rm Hom}\Bigl(\prod\limits_{i\geqslant 1} \Bbb Z,\Bbb Z\Bigr)$ has cardinality less than $2^{\mathfrak c}$?

I have read here that it has cardinality $\aleph_0$, which follows from a theorem of Specker, which I couldn’t find. I am looking for a less accurate bound to achieve the same conclusion, that $\prod\limits_{i\geqslant 1} \Bbb Z$ is not a free $\Bbb Z$-module. So for example $\leqslant \mathfrak c$ is good, or $<2^{\mathfrak c}$. […]

Generating connected module over a connected $K$-algebra

I have an algebraic question that I cannot solve. It is extracted from Adams and Margolis’ paper on modules over the Steenrod Algebra. Here is the problem : Let $K$ be a commutative ring with unit, $R$ a connected $K$-algebra (not commutative in my case), i.e., graded as $R \cong K \oplus R_1 \oplus R_2 […]

Show $M=0\iff M_{\mathfrak p}=0\iff M_{\mathfrak m}=0$.

Let $R$ a commutative ring and $M$ a $R-$module. Show that the following statement are equivalent: 1) $M=0$, 2) $M_{\mathfrak p}=0$ for all $\mathfrak p\in Spec(R)$, 3) $M_{\mathfrak m}=0$ for all $\mathfrak m\in Specm(R)$. Proof : $\bullet1)\Rightarrow 2)$: Let $M=0$. Let $\frac{a}{b}\in M_{\mathfrak p}$ (i.e. $b\notin \mathfrak p$). Then, $$\frac{a}{b}=\frac{a}{1}\frac{1}{b}=\frac{0}{1}$$ since $\frac{a}{1}=\frac{0}{1}$ and $\frac{0}{1}\frac{1}{b}=0$. $\bullet […]

$I\otimes I$ is torsion free for a principal ideal $I$ in domain $R$

Question is : Suppose $I$ is a principal ideal in a domain $R$. Prove that the $R$ module $I\otimes_R I$ is torsion free. Suppose we have $r(m\otimes n)=0$.. Just for simplicity assume that $m\otimes n$ is a simple tensor so we have $m,n\in I$. As $I$ is principal ideal we have $m=pa$ and $n=qa$ for […]

Are the Eisenstein integers the ring of integers of some algebraic number field? Can this be generalised?

The Eisenstein integers are $\mathbb Z[\omega]$ where $\omega$ is the primitive third root of unity. If $K$ is some algebraic number field, can $\mathcal O_K$ be isomorphic to $\mathbb Z[\omega]$? Attempt: The rank of $\mathcal O_K$ as a $\mathbb Z$-module is the degree of the extension $K$ of $\mathbb Q$. I believe that $\mathbb Z[\omega]$ […]