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Results tagged with modules
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user 84349
For questions on modules over rings.
19
votes
Accepted
When is $A\otimes R$ a free $R$-module?
Remark on projective modules over $\mathbb{Z}/n\mathbb{Z}$. Let $n$ be a positive integer. Then the following are equivalent:
The integer $n$ has no square factor. … Kaplansky, "Elementary divisors and modules", 1949.
[2] H. Matsumura, "Commutative Ring Theory", 1986. …
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8
votes
Accepted
Intersection of free/affine submodules, comparison with vector spaces
There is a natural generalization of the aforementioned dimension-based reasoning to modules $M$ over commutative domains or commutative Noetherian reduced rings. …
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8
votes
Accepted
$R$ a DVR with fraction field $K.$ What are the $R$-submodules of $K^n?$
The focus of [4] is the class of purely indecomposable modules (pi-modules), i.e., torsion-free indecomposable modules $M$ of finite rank with $p\text{-rank}(M) = 1$, or equivalently indecomposable pure … Rotman, "A note on completion of modules", 1960.
[2] D. Arnold, "A duality Torsion-free modules of finite rank over a discrete valuation ring", 1969.
[3] H. …
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8
votes
Accepted
When is countable direct-product of projective modules again projective ?
Under the assumption that a countable direct product of modules over $R$ means a direct product of countably many modules over $R$, I answer OP's question when $R$ is Noetherian. … Chase, "Direct product of modules", 1960.
[2] H. Bass, "Big projective modules are free", 1962.
[3] J. O'Neill, "When a ring is an $F$-ring", 1993. …
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7
votes
Accepted
Global to local principle for f.g. $\mathbb{Z}[x]$ modules
Your lemma easily follows from the Smith Normal Form Theorem, a result you already referred to. The short heuristic argument that you gave can indeed be turned into a short proof.
Nothing in the sequ …
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6
votes
Accepted
Modules over infinite rings which can not be a finite union of their proper submodules
Gottlieb, "Modules covered by finite unions of submodules", 1998.
Below lie the remains of my initial (awkward) answer.
This is only a long comment. …
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5
votes
Accepted
On integral domains over which special kind of modules are projective
An integral domain $R$ such that every proper non-zero $R$-submodule of $\text{Frac}(R)$ is projective is a local principal ideal ring. (The converse is David Handelman's comment above).
Indeed, we h …
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4
votes
Basis for free modules over an affine domain
, particularly in the resolution of Serre's problem on projective modules and its generalizations [9]. … Suslin, "A cancellation theorem for projective modules over algebras", 1977.
[5] H. Lindel, "On the Bass-Quillen conjecture concerning projective modules over polynomial rings", 1981.
[6] J. …
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2
votes
Accepted
Invariant Factors of a product $AU$ with $U$ an invertible matrix
I'll assume that elementary row and column transformations have determinant one. Otherwise the question is trivial, as darij grinberg pointed out in his comment above. My answer expands on Mohan's com …
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