Let $A[x]$ be the algebra of polynomials with coefficients in a $k$-algebra $A$. Assume that, for any simple $A[x]$-module $M$, one has $\text{End}_{A[x]}M = k$. Does it follow that any element of $J(A)$ is necessarily nilpotent?
2025-01-13 02:22:31.1736734951
Does it follow that any element of $J(A)$ is necessarily nilpotent?
131 Views Asked by user203482 https://math.techqa.club/user/user203482/detail AtRelated Questions in LINEAR-ALGEBRA
- Proving a set S is linearly dependent or independent
- An identity regarding linear operators and their adjoint between Hilbert spaces
- Show that $f(0)=f(-1)$ is a subspace
- Find the Jordan Normal From of a Matrix $A$
- Show CA=CB iff A=B
- Set of linear transformations which always produce a basis (generalising beyond $\mathbb{R}^2$)
- Linear Algebra minimal Polynomial
- Non-singularity of a matrix
- Finding a subspace such that a bilinear form is an inner product.
- Is the row space of a matrix (order n by m, m < n) of full column rank equal to $\mathbb{R}^m$?
Related Questions in ABSTRACT-ALGEBRA
- Projective Indecomposable modules of quiver algebra
- Binary relations for Cobb-Douglas
- Relations among these polynomials
- Number of necklaces of 16 beads with 8 red beads, 4 green beads and 4 yellow beads
- Page 99 of Hindry's Arithmetics, follows from exact sequence that $\text{N}(IJ) = \text{N}(J)\text{card}(J/IJ)$?
- How to write the identity permutation as a product of transpositions
- Is $H$ a subgroup?
- $x=(0,\overline{1})$ and $y=(0,\overline{2})$ generate the same ideal in $R=\mathbb{Z}\times\mathbb{Z}/5\mathbb{Z}$
- Having some problems with understanding conics and graphing (eccentricity)
- Is this Cayley Diagram contradictory?
Related Questions in POLYNOMIALS
- Relations among these polynomials
- If $f,g$ are non-zero polynomials and $f$ divides $g$, then $\partial f \leq \partial g$.
- If $z^5-32$ can be factorised into linear and quadratic factors over real coefficients as $(z^5-32)=(z-2)(z^2-pz+4)(z^2-qz+4)$,then find $p^2+2p.$
- All roots of the equation $a_0z^n+a_1z^{n-1}+.....+a_{n-1}z+a_n=n$,lie outside the circle with center at the origin and radius $\frac{n-1}{n}$.
- If the biquadratic $x^4+ax^3+bx^2+cx+d=0(a,b,c,d\in R)$ has $4$ non real roots,two with sum $3+4i$ and the other two with product $13+i$
- Pairwise Coprime Polynomials
- Surjective ring homomorphism from polynomial to complex numbers
- Fast polynomial division algorithm over finite field
- Find the polynomial of the fifth degree with real coefficients such that...
- Question about Polynomial and finding a model between two equation?
Related Questions in MODULES
- Proper essential extensions
- If module $M = N + mM$, then why is $m(M/N) = M/N$?
- On the invariance of basis number
- Geometric intuition for coherent rings, modules, and sheaves
- Finitely generated iff direct limits of subobjects are bounded by subobjects
- Finding highest weight of a $gl(3)$ submodule of a $gl(4)$-module
- A simple Congruence Equation
- On the $R$-linear independence of free $R$-module
- Do all submodules of $R^n$ have a complement in $R^n$?
- Formulas give irreducible representation, $SL(2, \mathbb{C})$.
Related Questions in ALGEBRAS
- Motivation for looking at the coalgebra structure of incidence algebra resp. group algebra
- Base change and power series rings
- Does it follow that any element of $J(A)$ is necessarily nilpotent?
- Automorphisms of the exceptional Jordan algebra preserve the determinant and trace.
- Showing bilinearity of the tensor product
- Can there be a set of numbers, which have properties like those of quaternions, but of dimension 3?
- One dimensional simple modules
- Augmented Algebras
- Does an algebra over a ring with unity have unity?
- Rings that are generated as an Algebra over a field by an arbitrary amount of algebraic elements
Trending Questions
- Induction on the number of equations
- How to convince a math teacher of this simple and obvious fact?
- Refuting the Anti-Cantor Cranks
- Find $E[XY|Y+Z=1 ]$
- 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?
- What are the Implications of having VΩ as a model for a theory?
- How do we know that the number $1$ is not equal to the number $-1$?
- Defining a Galois Field based on primitive element versus polynomial?
- Is computer science a branch of mathematics?
- Can't find the relationship between two columns of numbers. Please Help
- 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
- A community project: prove (or disprove) that $\sum_{n\geq 1}\frac{\sin(2^n)}{n}$ is convergent
- Alternative way of expressing a quantied statement with "Some"
Popular # Hahtags
real-analysis
calculus
linear-algebra
probability
abstract-algebra
integration
sequences-and-series
combinatorics
general-topology
matrices
functional-analysis
complex-analysis
geometry
group-theory
algebra-precalculus
probability-theory
ordinary-differential-equations
limits
analysis
number-theory
measure-theory
elementary-number-theory
statistics
multivariable-calculus
functions
derivatives
discrete-mathematics
differential-geometry
inequality
trigonometry
Popular Questions
- How many squares actually ARE in this picture? Is this a trick question with no right answer?
- What is the difference between independent and mutually exclusive events?
- Visually stunning math concepts which are easy to explain
- taylor series of $\ln(1+x)$?
- Determine if vectors are linearly independent
- What does it mean to have a determinant equal to zero?
- How to find mean and median from histogram
- Difference between "≈", "≃", and "≅"
- Easy way of memorizing values of sine, cosine, and tangent
- How to calculate the intersection of two planes?
- What does "∈" mean?
- If you roll a fair six sided die twice, what's the probability that you get the same number both times?
- Probability of getting exactly 2 heads in 3 coins tossed with order not important?
- Fourier transform for dummies
- Limit of $(1+ x/n)^n$ when $n$ tends to infinity