Let $f$ be module isomorphism $f:M_1 \rightarrow M_2$, with $M_1$ free. Then, is $M_2$ free too?
I tried to come up with a proof:
Let $B_1$ be the basis of $M_1$. Then we know that
$f(B_1)$ is linearly independent. $$f(m_1)=f(\sum r_ib_{1i})=\sum r_if(b_{1i})$$
for any $m_1 \in M_1$ and $b_{1i} \in B_1$.
Considering the fact that $f$ is bijective, is this enough to prove my question?
2026-03-25 14:21:42.1774448502
Do module isomorphisms preserve freeness?
107 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in CATEGORY-THEORY
- (From Awodey)$\sf C \cong D$ be equivalent categories then $\sf C$ has binary products if and only if $\sf D$ does.
- Continuous functor for a Grothendieck topology
- Showing that initial object is also terminal in preadditive category
- Is $ X \to \mathrm{CH}^i (X) $ covariant or contravariant?
- What concept does a natural transformation between two functors between two monoids viewed as categories correspond to?
- Please explain Mac Lane notation on page 48
- How do you prove that category of representations of $G_m$ is equivalent to the category of finite dimensional graded vector spaces?
- Terminal object for Prin(X,G) (principal $G$-bundles)
- Show that a functor which preserves colimits has a right adjoint
- Show that a certain functor preserves colimits and finite limits by verifying it on the stalks of sheaves
Related Questions in FREE-MODULES
- projective module which is a submodule of a finitely generated free module
- Flat modules over a PID
- On the generators of free $R$-modules.
- Is the linear dual of a cyclic $K[G]$-module $K[G]$-cyclic
- Is the quotient of a tensor product by a group a free module
- Cardinality of quotient group
- Why is a Nonzero Ideal of $\mathcal{O}_K$ Free of Rank $n$?
- Splitting two exact sequences in a compatible way
- When is every direct product of a ring also a free module?
- How does this commutative triangle arises in the idea of freely generated things?
Related Questions in MODULE-ISOMORPHISM
- Is the bidual of a C*-algebra isomorphic to the universal enveloping von Nemann algebra as a Banach algebra?
- $\ell(M/N) \leq \ell(M)$ if $M$ is an $A$-module
- Ring, but not a field.
- Are the $k[x]$-modules $M = k[x]/\langle x + 1 \rangle \oplus k[x]/\langle x + 1 \rangle $ and $N = k[x]/\langle (x + 1)^2 \rangle$ isomorphic?
- Do we have $R\simeq S$ for two submodules $R,S$ of $A^n$?
- Is $R$ finitely generated?
- FOL - If two models agree on every sentence are they isomorphic?
- Showing that $\frac{\mathbb{R}[x]}{\langle x \rangle}$ and $\frac{\mathbb{R}[x]}{\langle x-1 \rangle}$ are not isomorphic as $\mathbb{R}[x]$ modules.
- Are two submodules (where one is contained in the other) isomorphic if their quotientmodules are isomorphic?
- A problem about checking isomorphism of R-module
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?
The proof is correct. In general all properties of objects that make sense must be preserved by isomorphisms. Moreover, in categories that are not small there is a problem with defining equality of objects, since it must by definition the equality is a relation on the class of objects. Since a relation is a subset of the product, it is not well defined when objects form a proper class, in particular in the case of modules.