We know that every Artinian module is linearly compact. Is the converse true? Is this true that every finitely generated linearly compact module is Artinian? If not, is there example of a finitely generated module that be linearly compact but it is not Artinian? Thanks a lot.
2026-03-25 17:18:22.1774459102
Is every finitely generated linearly compact module Artinian?
82 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ALGEBRAIC-GEOMETRY
- How to see line bundle on $\mathbb P^1$ intuitively?
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- Is $ X \to \mathrm{CH}^i (X) $ covariant or contravariant?
- An irreducible $k$-scheme of finite type is "geometrically equidimensional".
- Global section of line bundle of degree 0
- Is there a variant of the implicit function theorem covering a branch of a curve around a singular point?
- Singular points of a curve
- Find Canonical equation of a Hyperbola
- Picard group of a fibration
- Finding a quartic with some prescribed multiplicities
Related Questions in COMMUTATIVE-ALGEBRA
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- Extending a linear action to monomials of higher degree
- Tensor product commutes with infinite products
- Example of simple modules
- Describe explicitly a minimal free resolution
- Ideals of $k[[x,y]]$
- $k[[x,y]]/I$ is a Gorenstein ring implies that $I$ is generated by 2 elements
- There is no ring map $\mathbb C[x] \to \mathbb C[x]$ swapping the prime ideals $(x-1)$ and $(x)$
- Inclusions in tensor products
- Principal Ideal Ring which is not Integral
Related Questions in COMPACTNESS
- Every nonempty perfect set in $\mathbb R^k$ is uncountable: Rudin's argument
- Help in understanding proof of Heine-Borel Theorem from Simmons
- Is the distance between those compact sets equal to $0$?
- Are compact groups acting on Polish spaces essentially Polish?
- Set of Positive Sequences that Sum to 1 is Compact under Product Topology?
- The space $D(A^\infty)$
- Proving the one-point compactification of a topological space is a topology
- Never Used Compact Closure...
- Continuity of the maximal element of a multi-valued function
- Consider the metric space of infinite sequences of 0s and 1s under this metric.
Related Questions in ARTINIAN
- $M$ finitely generated over $(R,\mathfrak{m})\Rightarrow M/\mathfrak{m}M$ artinian
- Primary ideals in an Artinial local ring
- Primary decomposition in a finite algebra
- Projectivity of a module
- Maximal and prime ideal in an artinian ring
- Example of a module that is finitely generated, finitely cogenerated and linearly compact, but not Artinian!
- If M is a a left module over $M_n(D)$ where $D$ is a division ring, M is Noetherian iff Artinian
- Checking if quotient ring is Noetherian or Artinian over a module
- Infinite Artin ring with only finitely many units
- Question about Hopkins-Levitzki Theorem's proof
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Find $E[XY|Y+Z=1 ]$
- Refuting the Anti-Cantor Cranks
- What are imaginary numbers?
- Determine the adjoint of $\tilde Q(x)$ for $\tilde Q(x)u:=(Qu)(x)$ where $Q:U→L^2(Ω,ℝ^d$ is a Hilbert-Schmidt operator and $U$ is a Hilbert space
- Why does this innovative method of subtraction from a third grader always work?
- How do we know that the number $1$ is not equal to the number $-1$?
- What are the Implications of having VΩ as a model for a theory?
- Defining a Galois Field based on primitive element versus polynomial?
- Can't find the relationship between two columns of numbers. Please Help
- Is computer science a branch of mathematics?
- Is there a bijection of $\mathbb{R}^n$ with itself such that the forward map is connected but the inverse is not?
- Identification of a quadrilateral as a trapezoid, rectangle, or square
- Generator of inertia group in function field extension
Popular # Hahtags
second-order-logic
numerical-methods
puzzle
logic
probability
number-theory
winding-number
real-analysis
integration
calculus
complex-analysis
sequences-and-series
proof-writing
set-theory
functions
homotopy-theory
elementary-number-theory
ordinary-differential-equations
circles
derivatives
game-theory
definite-integrals
elementary-set-theory
limits
multivariable-calculus
geometry
algebraic-number-theory
proof-verification
partial-derivative
algebra-precalculus
Popular Questions
- What is the integral of 1/x?
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- Is a matrix multiplied with its transpose something special?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- How to tell if a set of vectors spans a space?
- Calculus question taking derivative to find horizontal tangent line
- How to determine if a function is one-to-one?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- Is this Batman equation for real?
- How to find perpendicular vector to another vector?
- How to find mean and median from histogram
- How many sides does a circle have?
Any ring with a Morita duality is linearly compact in the discrete topology. See §24 and §30 in the book by Anderson-Fuller “Rings and Categories of Modules” for the definition.
It is a classical result by Müller that a ring with Morita duality is linearly compact in the discrete topology (B. J. Müller, Linear compactness and Morita duality, J. Algebra 16 (1970), pp. 60–66).
A simple example is the ring $F[[x]]$ of formal power series over a field $F$; see Example 10 in this paper by Weimin Xue (Bull. Austral. Math. Soc. 49 (1994), pp. 35–46). Such a ring is clearly not Artinian, but it is linearly compact in the discrete topology (and it is obviously finitely generated).