Commutative Algebra: Comprehensive Explanation

College
Algebra
Commutative Algebra

if $e_{1}, \dots, e_{n}$ is a complete set of orthogonal idempotents in a commutative ring, then $R = R e_{1} \times \dots \times R e_{n}$ is a direct product decomposition.
How can be proved this?

Dayanara Terry 2022-06-30

Suppose we're given a filtered algebra $A$ over a field $k$ with filtration $F_{∙} A$ over the subspaces of $A$ :
${0} \subseteq F_{0} A \subseteq \dots \subseteq F_{i} A \subseteq \dots \subseteq A,$
and suppose that
${g r}_{∙}^{F} A := ⨁_{i \in N_{0}} {g r}_{i}^{F} A$
is the associated graded algebra of $A$ , where ${g r}_{i}^{F} A := F_{i} A / F_{i - 1} A$ and ${g r}_{0}^{F} A = F_{0} A$ .
If ${g r}_{∙}^{F} A$ is commutative, does it follow that $F_{i + j} A \subseteq F_{i} A \cdot F_{j} A$ for all $i, j \in N_{0}$ ?

Cristopher Knox 2022-06-29

Let $A$ be the algebra given by the following multiplication table
$\begin{array}{cccc} \cdot & 0 & 1 & 2 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 1 \\ 2 & 0 & 0 & 1 & 2 \\ 3 & 0 & 1 & 2 & 3 \end{array}$
I need to prove that the variety generated by $A$ is exactly the variety of commutative semigroups satisfying $x^{3} \approx x^{4}$ .

Devin Anderson 2022-06-27

Let $(ℓ^{2}, {| | . ‖}_{2})$ with the coordinatewise product.Prove that $(ℓ^{2}, {| | . ‖}_{2})$ is a commutative and semisimple Banach algebra

Karina Trujillo 2022-06-27

Let $A$ be a ring and $X = s p e c (A)$ , the prime spectrum of $A$ . Prove that $X$ is quasi-compact.
Definition of quasi compact: each open covering of $X$ has a finite subcovering of $X$ .
It is enough to considering the covering in the basis ${X_{f} | f \in A}$ .
Let the set of ${X_{f_{i}} | f_{i} \in A, i \in I}$ . I is some index set. It is obvious that $f_{i}$ with $i \in I$ generates the unit ideal. But why there exists a finite subset $J$ of $I$ such that $f_{i}$ with $i \in J$ generates the unit ideal?

Kiana Dodson 2022-06-26

Let $A$ and $B$ both be associative, commutative, finite-dimensional, complex algebras of dimension $n$ , e.g. over $C$ .
What are some necessary and sufficient conditions for $A$ and $B$ to be isomorphic?
In particular, I'm interested in the situation where $A$ and $B$ have zero divisors, but both have no nilpotent elements. Is this sufficient to declare $A$ and $B$ isomorphic?
Two examples of such algebras are the pointwise multiplication on $C^{n}$ , and the convolution algebra on $C^{n}$ . These are isomorphic via the discrete Fourier transform. Are all complex algebras of the same dimension, which don't have nilpotent elements, isomorphic to these two?

Yahir Tucker 2022-06-26

In a commutative ring with 1, every proper ideal is contained in a maximal ideal.
and we prove it using Zorn's lemma, that is, $I$ is an ideal, $P = {I \subset A ∣ A is an ideal}$ , then by set inclusion, every totally ordered subset has a bound, then $P$ has a maximal element $M$ .
My question is why $M$ must contain $I$ ?

tr2os8x 2022-06-26

Let $A$ be a finite dimensional separable commutattive algebra over a field $k$ . The book A Course in Computational Algebraic Number Theory by Cohen says that $A$ has a single generator as an algebra over $k$ . I tried to prove this but failed. $A$ is a finite product of separable extension fields $K_{i}$ over $k$ . It is well known that each $K_{i}$ has a single generator over $k$ , but I don't know how to use this fact to prove it.

Yahir Crane 2022-06-26

Let $V$ be a vector space over a field $F$ and for all $(i, j) \in E_{n}^{2}$ ( $E_{n} = {1, 2, . . ., n}$ ), $P_{i j} : V \to V$ is a linear map. Suppose moreover that...
1) $\forall (i_{1}, j_{1}, i_{2}, j_{2}) \in E_{n}^{4}$
2) ${v_{j} \in V}_{1 \leq j \leq n}$ obey the following set of equations
$\forall i \in E_{n} : \sum_{j = 1}^{n} P_{i j} v_{j} = 0.$
Then $\forall j \in E_{n}$
$(det P) v_{j} = 0$
where $P$ is the $n \times n$ matrix with entries $P_{i j}$ and $det P$ has the same combinatorial structure as the usual determinant, yet with all multiplications replaced by map compositions.
Question: What name(s) is given to this type of result in the mathematical literature? Are there short, elegant proofs?

Gabriella Sellers 2022-06-25

Are all algebraic commutative operations always associative?
I know that there are many algebraic associative operations which are commutative and which are not commutative.
for example multiplications of matrices as associative operation is not commutative.

preityk7t 2022-06-25

Does there exist a non-commutative algebraic structure with the following properties?
1. $| M | \geq 2$
2. For all $a$ , $b$ , $c$ $\in M$ , $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ .
3. For all $a$ , $b$ $\in M$ with a≠b, exactly one of the equations $a \cdot x = b$ and $b \cdot x = a$ has a solution for $x$ in $M$ .
4. For all $a$ , $b$ $\in M$ , the equation $a \cdot x = b$ has a solution for $x$ in $M$ if and only if the equation $y \cdot a = b$ has a solution for $y$ in $M$ .

aligass2004yi 2022-06-25

According to some lecture notes I am reading, "it is not so difficult to find an example of a commutative unital Banach algebra which is not isomorphic to $C (X)$ for some compact Hausdorff space $X$ "...
Well I was thinking about using the disk algebra $A$ of continuous functions on the disk $D = {z ∣ | z | \leq 1}$ which are holomorphic on the interior of $D$ , with the sup norm.
This should be a commutative unital Banach algebra. However, since it is the beginning of the chapter about the Gelfand transformation, I would like to prove that $A$ is not isomorphic to $C (X)$ without using Gelfand's result.
What other tools could I use to prove this fact?

vittorecostao1 2022-06-24

Is it true that for simple $C^{*}$ -algebras, meaning that they don't have non-trivial two-sided ideals, it holds that they are necessarily non-commutative? And why?

Summer Bradford 2022-06-23

The statement-title above is a result that should follow from the hint I got to solve it: "Show that the inversion map $i : G \to G, g ⟼ g^{- 1}$ is a Lie group homomorphism if $G$ is commutative, and then study the resulting Lie algebra homomorphism." The first part of this hint was easy, but I am not getting any further. I think the corresponding homomorphism should be the total derivative of i evaluated in the identity of $G$ , but I'm not getting anywhere with this idear yet.
I think the trivial Lie bracket should be $[X, Y] = 0$ for two vector fields on $G$ , which holds iff the corresponding flows of $X$ and $Y$ commute.

Extrakt04 2022-06-21

I know that all the non-unital commutative $C^{*}$ algebras are isomorphic to $C_{0} (Ω)$ ,where $Ω$ is a locally compact space. Can anyone show me some common non-unital commutative examples.I only know the $c_{0}$ space.

sviraju6d 2022-06-21

Is there an algebraic way to construct the coproduct, as for example some form of topological tensor product or similar?
Let $A$ be a commutative unital C*-Algebra and let $X = S p e c (A)$ be the corresponding compact Hausdorff space of characters. By Gelfand-Naimark duality we know that
$X \times X = S p e c (A ∐ A)$
or in other words
$A ∐ A = C (X \times X) .$

gnatopoditw 2022-06-20

One forms $Ω^{1} (M)$ is a module over $C^{\infty} (M)$ , therefore does that make $C^{\infty} (M)$ an algebra or a commutative ring?

Boilanubjaini8f 2022-06-20

Let $A$ be some $F$ -Algebra, for some field $F$ , with the property that $A$ is finite dimensional over $F$ . Is $A$ always commutative?

Emanuel Keith 2022-06-20

Spectral radius in Banach algebra is commutative
I want to show that for a Banach algebra $A$ and elements $x, y \in A$ , we have
$r_{A} (x y) = r_{A} (y x),$
where $r_{A}$ is a spectral radius. This is how I am trying to do that:
$r_{a} (x y) = lim_{n \to \infty} ‖ (x y)^{n} ‖^{\frac{1}{n}} = lim_{n \to \infty} ‖ x (y x)^{n - 1} y ‖^{\frac{1}{n}} .$
And this is where I stuck, since I only know that $‖ x y ‖ ⩽ ‖ x ‖ ‖ y ‖$ . Could you please suggest any ideas on how to proceed with the proof?

kixEffinsoj 2022-06-20

Let $A$ be a unital not-necessarily commutative algebra, defined over $R$ or $C$ . Take some $α$ a non-unital algebra automorphism of $A$ . Is it possible to find an example for $A$ and $α$ such that $1 - α (1)$ is non-invertible in $A$ ?

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

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