# Tagged Questions

For questions about modules over rings, concerning either their properties in general or regarding specific cases.

52 views

10 views

### Submodules of direct sum of simple modules [duplicate]

Is it true that if $N$ is a simple module, then the only non trivial submodule of $N\oplus N$ is $N$? I am aware of the "diagonal" submodule $\{(x,x)\mid x\in N\}$, but that is isomorphic to $N$?
35 views

### Abelian group structure and structure of Z[i]

Let $F = \Bbb{Z_p}$. For which prime integers $p$ does the additive group $F^1$ have a structure of $\Bbb{Z}[i]$-module? How about for $F^2$? I'm not really sure on how to approach this question, do ...
74 views

41 views

### Module over an infinite dimensional algebra

I have two question related to infinite dimensional algebra I have been seen a lot of example about Module over a finite dimensional $k-$algebra, but I could not find a literature about Module ...
29 views

### Dimension of a basis for a module using the quotient map

I'm going through a proof that the size of a basis for a module is uniquely determined and can't see why the following line from my lecture notes follows: "We now claim that if $X$ is a basis for ...
212 views

### Computing (the ring structure of) $\mathrm{Ext}^\bullet_R(k,k)$ for $R=k[x]/(x^2)$

Let $k$ be some field (say of characteristic zero, if it matters) and define $$R=k[x]/(x^2).$$ I want to compute $$\mathrm{Ext}^\bullet_R(k,k)$$ and, in particular, the ring structure on it (though I ...
51 views

21 views

### two sided ideals and idempontents

Let I be a two sided ideal of R. Prove that I=eR for some central idempotent e ϵ R if and only if R=I+J for some two sided ideal J. When this occurs, show that e and J are uniquely determined by I. ...
24 views

### Homomorphism of Matrices over Extended Field

Let $E$ and $F$ be fields such that $E$ is an extension field of $F$ and $[E:F] = n$. Let $M_k(F)$ denote the ring of $k \times k$ matrices over $F$. Does there exists a homomorphism from $M_k(E)$ to ...
37 views

### Multiplication map for algebras in the sense of Hopf algebra

Let $R$ be a commutative ring with unity and $A$ a ring with unity. In the definition of algebras, we require a multiplication map $\mu: A\otimes A \rightarrow A$ satisfying certain property. Indeed, ...
58 views

41 views

### Projective Dimension and Schanuel's Lemma

Let $R$ be a ring and $M$ a (say, left) $R$-module of projective dimension $n$. According to Noncommutative Noetherian Rings, any projective resolution of $M$ can be terminated at length $n$, and this ...
16 views

### Why is the minimal polynomial of linear transformation $T$ the lcm of all elementary divisor

Suppose you have a vector space $V$ over $K$ and you give $V$ a $K[x]$ module structure via a linear operator $T$. Why then is the minimal polynomial of $T$ necessarily the lowest common multiple of ...
16 views

### Equivalence of definitions of minimal polynomial of a linear transformation $T$ over a vector space

Let $V$ be a vector space over $K$ and let $T\in End_{K}(V)$. Then $V$ has a $K[x]$-module structure via the operation defined as $f(x)v=f(T(v))$. Using this we can know a lot about the structure of ...
22 views

### Exterior power of a free module [duplicate]

Suppose $M$ is a free R module where $R$ is commutative and unital. Is $\Lambda^n M$, the nth exterior power of $M$ free?