Commutative Algebra: Comprehensive Explanation

College
Algebra
Commutative Algebra

Question: What's the cleanest proof that $H^{} (S y m (V)) ≅ S y m (H^{} (V))$ ?

hovudverkocym6 2022-05-12

Suppose that $A$ is a natural Banach function algebra on $K$ , a compact Hausdorff space. So $A$ is realised as an algebra of continuous functions on $K$ , is a Banach algebra for some norm (necessarily dominating the supremum norm) and each character on $A$ is given by evaluation at a point of $K$ .
If $F \subseteq K$ is closed, then
$I (F) = {f \in A : f (k) = 0 (k \in F)}$
is a closed ideal in $A$ . If e.g. $A = C (K)$ then every closed ideal is of this form.What's a simple example of an $A$ where not every closed ideal is of this form?
Can we find an $A$ which is conjugate closed?

bedblogi38am 2022-05-11

Let x not be quasinilpotent, so $l i m_{n \to \infty} | | x^{n} | |^{1 / n} = λ \neq 0$ . Let $π (x) = \bar{x}$ , where $π (x) : A \to A / r a d (A)$ is the canonical quotient map. Suppose that $| | {\bar{x}}^{n} | |^{1 / n} = 0$ . Then it's spectrum $σ (\bar{x}) = 0$ , so $\bar{x}$ is not invertible in $A / r a d (A)$ , and thus generates a proper ideal in $A / r a d (A)$ . So then $π^{- 1} (\bar{x})$ generates a proper ideal in $A$ containing $r a d (A)$ .
From here, if $r a d (A)$ is maximal I think I'd have a contradiction, but I don't know if that's true. If not, does anyone have another strategy I could try?

Write an algebraic expression for the following: 6 more than a number c. PLEASEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEEE

lasquiyas5loaa 2022-04-12

I see the following notation: for a ring $A,$ a prime ideal $q,$ , and $x \in A$ .
$(q : x) = (1) .$
The question is: what does $(q : x)$ mean? The ideal generated by $q, x ?$ Then why do we need to use a colon?

vilitatelp014 2022-04-10

An associative magma is said to be a semigroup. A semigroup with identity is said to be a monoid... Is there a name for an associative and commutative algebra?

Define adjoint of operators

Marla Payton 2022-01-07

Show that $L^{1} (R)$ a Banach algebra Commutative

Brock Brown 2022-01-07

Let * be a binary operation on the set of real numbers R defined as follows:
$a \cdot b = a + b - 3 {(a b)}^{2}$ , where $a, b \in R$
- Prove that * is commutative but not associative algebraic operation on R.
- Find the identity element for * .
- Show that 1 has two inverses with respect to *.

obrozenecy6 2022-01-06

Say a function is commutative if it remains unchanged under any permutation of its arguments. E.g. $f (0, 1) = f (1, 0)$ . (Alternatively we could describe these as functions over multi-sets, or say that they are reflective about any hyperplane $x_{i} = x_{j}$ ). Some examples are sum, product and average.
1) Is there a name for these functions? Google searches for commutative and reflective functions dont

killjoy1990xb9 2022-01-06

Consider the algebraic structure $(Z \times Z, ⊥)$ , with $(a, b) ⊥ (c, d) = (a + c, b d)$ for each $a, b, c, d \in Z$ .
Prove that $(Z \times Z, ⊥)$ is a commutative monoid

socorraob 2022-01-06

What is meant by the term commutative in algebra? Explain along with an example.

Carol Valentine 2022-01-06

We have a result if R is commutative, then so is any quotient ring R/I, for any ideal I. If we take the contrapositive, we see that if R/I is non commutative, then R is non commutative. Is there any noncommutative ring that has a commutative quotient ring?

guringpw 2022-01-04

Algebraic results with easier non-commutative proofs than commutative proofs
(1) The dimension of a vector space over a field and the rank of a free module over a division ring.
(2) Theorem for $C [x, y]$ and Theorem for the first Weyl algebra (it is not so difficult to show that in the commutative result we can replace C by any field of characteristic zero).

zakinutuzi 2022-01-04

Is a direct sum of two commutative rings still commutative? suppose f is in F, g is in G. why fg=gf?

agreseza 2022-01-04

If A is a commutative C*-subalgebra of linear bounded operator space B(H) on some Hilbert space H, so is the double commutant A. It follows from A is dense in A and the multiplication is continuous on each factor respectively, respect to the strong operator topology.
1) Can the assertion A is commutative be verified in an algebraic way?
2) Or, a more general question: If A is a commutative subalgebra of an algebra B, is the double commutant A of A also commutative?

Linda Seales 2022-01-04

As I was working through an algebra textbook, I noticed that a field A is a commutative ring. But is it possible for $A \subset B \subset C$ where $A, C$ are fields and $B$ is not?

untchick04tm 2022-01-04

Prove, that the vector Space Hat (n; F) with the multipliсation
$A \cdot B = A B - B A$ is a F-algebra (algebra over a field F) is such an algbera associative, commutative, untiary?

Tara Alvarado 2022-01-04

Solve following system of equations of non-negative integers using methods of Commutative Algebra
$3 σ_{1} + 2 σ_{2} + σ_{3} + 2 σ_{4} = 10$
$4 σ_{1} + 3 σ_{2} + σ_{3} = 12$
$2 σ_{1} + 4 σ_{2} + 2 σ_{3} + σ_{4} = 25$

In basic terms, commutative algebra represents a complex study of the rings that take place in algebraic number theory. It also relates to solving problems and questions based on algebraic geometry. You will see that there are solutions related to Dedekind rings and various commutative patterns. Commutative algebra solutions will be quite short in most cases, yet one should start with examples that are represented and link them with various questions. Take your time to post your commutative algebra example problem as well and always compare your question to similar questions as it will help you provide the required data. When you are dealing with inequality and graph solution tasks as you are seeking commutative equ

Commutative Algebra

Commutative Algebra Explained: In-Depth Examples, Equations, and Expert Insights