Given that $R$ is a commutative unitary ring and $I$ and $J$ are two ideals of $R$ such that $R=I\oplus J$ (as an internal direct sum), how do you show that there exists two idempotent elements $e$ and $f$ in $R$ such that $I= Re$ and $J= Rf$?
2026-03-26 05:55:54.1774504554
For $R=I\oplus J$, prove that $I= Re$ and $J= Rf$ where $e$ and $f$ are two idempotent elements
256 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 RING-THEORY
- Jacobson radical = nilradical iff every open set of $\text{Spec}A$ contains a closed point.
- A commutative ring is prime if and only if it is a domain.
- Find gcd and invertible elements of a ring.
- Prove that $R[x]$ is an integral domain if and only if $R$ is an integral domain.
- Prove that $Z[i]/(5)$ is not a field. Check proof?
- If $P$ is a prime ideal of $R[x;\delta]$ such as $P\cap R=\{0\}$, is $P(Q[x;\delta])$ also prime?
- Let $R$ be a simple ring having a minimal left ideal $L$. Then every simple $R$-module is isomorphic to $L$.
- A quotient of a polynomial ring
- Does a ring isomorphism between two $F$-algebras must be a $F$-linear transformation
- Prove that a ring of fractions is a local ring
Related Questions in IDEMPOTENTS
- Prove that an idempotent element must be either 0, 1 or a zero-divisor.
- What is the set {$e\in(R/ I)\times(R/J): e$ is idempotent}
- The idempotent elements of Eisenstein Integers
- Prove that $A-I_n$ is idempotent
- Idempotent substitution $\theta$
- Relations of structures related to conjugate idempotents
- Composition series of regular module
- Is $A^3=A$ a condition for idempotency of matrices?
- Is it true that $X(X'X)^{-1}X'-J/n$ is idempotent, where $J$ is an $n$ by $n$ matrix of ones?
- Idempotents over a ring with zero divisors
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?
Try writing the identity of $R$ as an element of $I$ plus an element of $J$ .