Given the language $\{+,0\}$, I would like to prove that $\mathbb{Z} \oplus \mathbb{Z}$ is not isomorphic to $\mathbb{Z}$, but I am not sure where to start?
2026-03-29 15:21:31.1774797691
Isomorphism between direct sum of $\mathbb{Z}$ and $\mathbb{Z}$
1.5k Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
Related Questions in ABSTRACT-ALGEBRA
- Feel lost in the scheme of the reducibility of polynomials over $\Bbb Z$ or $\Bbb Q$
- Integral Domain and Degree of Polynomials in $R[X]$
- Fixed points of automorphisms of $\mathbb{Q}(\zeta)$
- Group with order $pq$ has subgroups of order $p$ and $q$
- A commutative ring is prime if and only if it is a domain.
- Conjugacy class formula
- Find gcd and invertible elements of a ring.
- Extending a linear action to monomials of higher degree
- polynomial remainder theorem proof, is it legit?
- $(2,1+\sqrt{-5}) \not \cong \mathbb{Z}[\sqrt{-5}]$ as $\mathbb{Z}[\sqrt{-5}]$-module
Related Questions in DIRECT-SUM
- Finding subspaces with trivial intersection
- Direct sum and the inclusion property
- direct sum of injective hull of two modules is equal to the injective hull of direct sum of those modules
- What does a direct sum of tensor products look like?
- does the direct sum of constant sequences and null sequences gives convergent sequence Vector space
- Existence of Subspace so direct sum gives the orignal vector space.
- A matrix has $n$ independent eigenvectors $\Rightarrow\Bbb R^n$ is the direct sum of the eigenspaces
- $\dim(\mathbb{V}_1 \oplus ...\oplus \mathbb{V}_k) = \dim\mathbb{V}_1+...+\dim\mathbb{V}_k$
- Product/coproduct properties: If $N_1\simeq N_2$ in some category, then $N_1\times N_3\simeq N_2\times N_3$?
- Direct Sums of Abelian Groups/$R$-Modules
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?
HINT: Let $\varphi(x)$ be the formula $\exists y\,(y+y=x)$, and consider the sentence
$$\forall x\forall y\Big(\neg\varphi(x)\land\neg\varphi(y)\to\varphi(x+y)\Big)\;.$$