A ring is known to be left p.p. if every principal left ideal is projective. It is well known that this condition is equivalent to the fact that every annihilator of each element is generated by an idempotent. However, I cannot easily see how this works. Can anyone give me at least a hint? It will be a great help for me.
2026-03-26 22:15:10.1774563310
On the equivalence condition of p.p. rings
48 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
There are 1 best solutions below
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 PROJECTIVE-MODULE
- $R$ a domain subset of a field $K$. $I\trianglelefteq R$, show $I$ is a projective $R$-module
- projective module which is a submodule of a finitely generated free module
- Flat modules over a PID
- Projectivity of a module
- Proving a module is projective.
- Proving a specific module is projective.
- $M\otimes_A N\cong A$ implies $M$ is left $A$-projective?
- Proof of equivalence of two definitions of a projective module
- Problem based on Projective and Injective Module
- Splitting two exact sequences in a compatible way
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?
Hint: The sequence $$ 0\rightarrow Ann(r)\rightarrow R \rightarrow Rr\rightarrow 0 $$ splits, hence every such anihilator $Ann(r)$ is a summand of $R$, and hence has the form $Re$ for an idempotent $e$ in $R$. A possible reference is this book on ring theory. More references on PP-rings and Baer rings, see here.