# Tagged Questions

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### Module over an associative ring with unity and axioms of projective geometry

According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are: G1: Every line contains at least ...
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### Endomorphism rings of a $k$-algebra

Let $k$ be a field and $R$ be the $3$-dimensional $k$-algebra $\left [\begin{array}\ k & k \\ 0 & k \end{array} \right ]$. I have two conjectures: (1) Is any simple right $R$-module has ...
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### Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
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### Structure theorem of modules in the graded case

I ran into an exercise in D. Passman's book A course in ring theory (page 130) asking me to prove the structure theorem of modules over PIDs in the graded case. Could anyone point me in the right ...
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### Contrasting definitions of bimodules? An illusion?

Recently my definition of a bimodule over a $k$-algebra has been challenged and I believe both definitions to be equivalent, am I wrong? Notation: $k$ is a commutative ring and $A$ is a (unital ...
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### Any module is the colimit of its finitely generated submodules.

Fact A : If $E_i$ is a directed system of $R$-modules, and $F$ is another $R$-module. Then $$\varinjlim(F\otimes E_{i})=F\otimes(\varinjlim E_{i}).$$ Fact B : "Every module is a direct limit of its ...
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### Question about Universal Mapping Property

Now I seem to understand the construction of the tensor product of $R$-Modules by defining $$F_R(M \times N) := \bigoplus_{(m,n) \in M \times N} R\delta_{(m,n)}$$ and constructing the submodule $D$ in ...
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### Tensor product of free modules over free algebra

Suppose $M$ and $N$ are modules over a (commutative, unital) ring $S$. Let $R$ be a subring of $S$ such that $S,M$ and $N$ are all free, finitely generated modules over $R$. Question: Under what ...
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### How are 'irreducible' elements called in module theory?

Let $R$ be a 'nice' ring (whatever that means, I guess commutative with unity $1_R$) and let $M$ be an $R$--module. Is there a name for the following property? An element $m \in M$ is called ... iff. ...
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### Graded rings and modules - which book to refer to?

Could you recommend a book with a good treatment of the theory of graded rings and modules?
The Statement I suspect the following proposition is well known, but I found no reference. Proposition If $A$ is a principal ideal domain, if $I$ is a nonzero ideal of $A$, and if $M$ is an ...
I have the following problem Is the product of divisible $R$-modules divisible? I think it is not. But I need some counterexample to this, somebody can give me a "place" to search one? Thanks a ...