Science topic

# Commutative Algebra - Science topic

Explore the latest questions and answers in Commutative Algebra, and find Commutative Algebra experts.

Questions related to Commutative Algebra

other methods apart from commutative algebra, minimum number of colors, relationship between ideals with edges and nodes,

The unexploited unification of general relativity and quantum physics is a painstaking issue. Is it feasible to build a nonempty set, with a binary operation defined on it, that encompasses both the theories as subsets, making it possible to join together two of their most dissimilar marks, i.e., the commutativity detectable in our macroscopic relativistic world and the non-commutativity detectable in the quantum, microscopic world? Could the gravitational field be the physical counterpart able to throw a bridge between relativity and quantum mechanics? It is feasible that gravity stands for an operator able to reduce the countless orthonormal bases required by quantum mechanics to just one, i.e., the relativistic basis of an observer located in a single cosmic area?

What do you think?

- We know ,euclidean algorthm is feasible in the set of integers.
- Taking motivation from this,we define an Euclidean Domain(E.D.) as follows:

R is an E.D. if it is a domain,andwe have a map d:R->non-zero integers,

such that, for given a,b(non-zero),

there exists q,r with

a=bq+r and d(q)<d(r).

- Now,this d function ,for any abstract ED,is the counterpart of |.| function,in case of intgers,

we have ,a=bq+r with,0 \le r < |b|.

*Now the question is:*

*[] It turns out (q,r) that exists for a given (a,b) in case of intgers ,is unique .**[A simple proof would be:if another (q',r') exists for same (a,b),*

*if q'=q,we have r'=r,and we are done.*

*and if,q' and q are distinct,bq+r=bq'+r' implies r' and r differ by multiplies of b,thus if*0 \le r < |b| holds,it is clear it won't hold for r',and thus we can never have (q',r') and (q,r) distinct,and we are done.]

*[] But,one now would ask,is it true as well that for given (a,b) in any E.D. the existing (q,r) would be unique ? If yes,we need a proof,and clearly,the same proof does not work,as d(.) is much more generalized than |.| .Or,we need a counter eg!*

*__________________________________________________________________________________*In the definition of a group, several authors include the Closure Axiom but several others drop it. What is the real picture? Does the Closure Axiom still have importance once it is given that 'o' is a binary operation on the set G?

I have attached my definition and some results about this.

What are the current trends in Commutative Algebra and its interactions with Algebraic Geometry? What are the important articles that one should read for starting to research in this area?

I'm currently studying in mathematics major, undergraduate program and doing my very first scientific writing and research. The topic that I get is about "Generalized power series rings", but I still have have no firm understanding about it. I've tried to look some journals in research gate and save some of it.

Could you recommend me books or scientific journals for me to read so that I could have more in-depth understanding on the topic?

Any recommendations is highly appreciated and welcome,Thank you.

What are the conditions for a commutative ring with unit R to verify the proprety: every locally cyclic R-module is cyclic ?

I recall that an R-module is cyclic if it is generated by one element.

And is locally cyclic if every localization by a prime ideal will be generated by one element as a R_p-module.

This is obvious when R is local, but whzt are the general conditions ?

I am trying to understand automorphism group of zerodivisor graph. If R is an infinite Noetherian ring, The automorphism group of corresponding zero-divisor graph is infinite. Do there exist any infinite non-Noetherian ring such that the automorphism group of corresponding zero-divisor graph is finite.

To make two square matrixs

**A**and**B**satisfy the commutative law of multiplication, say,**AB**=**BA**, at least one of the following conditions is required:Condition 1:

**A**=**B**.Condition 2: A=k

**I**or B=k**I**.**I**refers to the Identity matrix.Condition 3: Both A and B are diagonal matrices.

Any other conditions?

hi

i want to know about endomorphism ring of power series rings. if anyone know about that or road a paper about that please share it with me.

thank you

Starting in polynomial extents, computational commutative algebra, branched into applications via Gröbner basis theory and the generalized concept of approximate commutative algebra. But what is next? Do you believe the computation extends to non-commutative algebra?

I have that R is the k-algebra (k is a field) finitely generated by S={f1,...,fm}⊂k[x1,⋯,xn] and this set of polynomials is minimal with respect to inclusion (i.e., e do not have redundant elements). However, I know that the fi's are algebraically dependent. Can be the number m=|S| unique? What are the conditions?

Let R be a commutative ring with unity. Given two algebras A and A' over a field K, both with R-grading form. Prove that B:= A \otimes_R A' (the tensor product of A and A') has a R²-grading.

Any tip of how to prove this?

Suppose there is a polynomial

*P*(*x*) of degree ≤*n*(and in characteristic zero). Then, if*P*(*x*) takes value 0 for*n*+1 different values of*x*, then all its coefficients are actually zeros.Let, however, all these coefficients be integer, and each of them be not greater than 10 in absolute value. Then, one single condition

*P*(11)=0 is clearly enough to guarantee that*P*(*x*) is identical zero.Then it may be asked about similar results for an algebraic function of several variables, instead of our simple

*P*(*x*). Most likely, there is already a big theory about such things, so why not learn it. The question is where I can find the best exposition of such theory.Let (R,m) be a Cohen-Macaulay local ring, with canonical module ω, and let I be an ideal of R.

Question: Is the minimal number of generators of $Hom_R(ω, Iω)$ equal to the minimal number of generators of I?

The answer is affirmative if R is Gorenstein.

Does anybody know an answer in the general case, please?

Thank you so much.

Let J be an ideal in a polynomial ring R[x] and let p(x) be a monic non-constant polynomial, where R is a commutative Noetherian ring with identity. Let J(p) denote the ideal of R[x] generated by the polynomials f(p(x)), where f ranges over the elements of J. Clearly J(p) cannot require more generators than J. Could J(p) require fewer generators than J? I attached a partial attempt at a proof.

see p. 333 of the attached paper.

that is :

How to find formula formula for steady temperatures i

**n thin semi-infinite plate****y > 0**whose faces are insulated and whose edge**y = 0**is kept at temperature zero except for the partion**-1<x<1,****y=0**where it is kept at temperature unity. ?We have “big” and “small” in our science and the related numerical cognizing way to universal things around us （one of the mathematical cognizing ways）. So, we have “very big (such as 1000

^{1000})” and “very small (such as 1/1000^{1000)}”, “extremely big” and “extremely small”… in mathematics. When “infinite” came into our science and mathematics, we naturally and logically have “infinite big (infinities)” and “infinite small (infinitesimals)”.Many people think we really have had many different mathematical definitions (given by Cantor) for infinite: infinity (infinite big) in set theory. So, when talk about the mathematical definition for “infinite”, people only think about “infinite big” but negate “infinite small”.

Should the mathematical definition for “infinite” cover both “infinite big (infinities)” and “infinite small (infinitesimals)” or

**?***only for the half: infinite big (infinities)*Hello,

Please see attached file.

Is it interresting mathematically speaking ?

Could we desambiguous more economically the grouping of moves notation by another operation than addition ? Is it inspiring to think about a pattern symbolic one ?

I need examples of

(1) Two finitely generated ideals in a ring R such that their intersection is not finitely generated and

(2) An element x in a ring R such that (0:x) is not finitely generated.

Let P=k[x1...xn] be a poly over a field. Suppose that R=k[x1...xn]/I. The canonical module of R is $ω_R=Ext^{n−dim(R)}_P(R,P)$.

The question is that is there any upper bound for the min num of generators of ωR in terms of r=the min num of gen of I? and\or the embedding dim of R?

One may add more reasonable conditions to R, e.g. assume that R is Cohen-Macaulay?

and\or

Assume that I is almost complete intersection i.e. r=codim(I)+1.

A desired upper bound would be (r \choose 2) i.e. μ(ω_R)≤(r\choose 2).

Recall that min num gen ωR denoted by μ(ωR) is also called the type of the ring R denoted by r(R). And the latter is one iff R is Gorenstein , this is a result of P.Roberts.

Let's also remind that in the CM case the question is tantamount to ask an upper bound for the last betti number in the min free resolution of R over P.

Considering G a group and H its subgroup. <br />F a field, and F(G) a group ring. Since F(G) is a ring, we can consider also that F(H) is a subring, and then F(G) as a left or right F(H)-module.<br /><br /><strong>*</strong>I thought about writting an isomorphism between F(G)/F(H) and G/H. If one is finitely generated, I could say rank[F(G)/F(H)] = rank[G/H] = [G : H].<br />I am not sure about this proposal.

Knowing that R is a commutative ring and R[X] is a free R-module.

It seems obvious because any R-module can be written as quotient of a free R-module and any of its submodules.

As R[X²] is a submodule of R[X] over R, then it is an ideal of R.

It turn to simply show a bijective homomorphism, but I encounter problems describing the elements in this relations.

Let A be a commutative nil-algebra of nilindex n and dimension finite over an algebraically closed field of characteristic not 2, such that there exist a non zero ideal I such that I=AI. (but I \cdot I \neq I. because in this case, Dr Misha constructed a counterexample)

Then Null(A) \cap A^2 = {0}.

By A a commutative finite-dimensional algebra means a vector space of dimension finite over a field k with a generic simetric bilinear operator. In this particular case, additionally the product satisfy for every x \in A x^2 x^2= x^4. (Commutative power-associative algebras).

By nil-algebra of nilindex n means that for every a \in A, and every arrange of n products of a, this product is zero.

This particular kind of algebras are not associative, and don' t have 1.

By Null(A) means the set of elements a \in A such that the right multiplication map R_a (or equivalent, the left map L_a) is the zero map.

Thanks everybody, and I'm sorry for all my mistakes.

I'm searching for a counterexample for these sentence " every nonessential ideal in C(X) is not necessarily a z-ideal"

This is a part of an equivalent statements theorem in Gillman and Jerison, Rings of continuous functions.

May anybody help me in proving it?

I am trying to find an example of IF ring R (every injective R-module is flat) which is not von Neumann regular (every R-module is flat).

Suppose I have an affine algebraic variety, and I suspect that a certain element of its coordinate ring can be represented as a product of two elements having some nice form. Are there computer algorithms for such factoring?

Let there be a ring

*r*of polynomials in six indeterminates*t,u,v,w,x,y*with complex numbers as coefficients. Take then a quadratic extension (if this is the right word)*R*of*r*by adding a new letter*z*which is the square root of a polynomial*p*in our six indeterminates (these latter are, of course, transcendental over**C**). Polynomial*p*is homogeneous of 14th degree, if this matters. The question is: where can I read about (algorithms of) factorizing elements in*R*? Answer for only homogeneous elements will suffice (assuming*z*has degree 7), but if a factorization is not unique, I want all of them!Remark. Straightforward attempts using primary decomposition algorithm in Singular proved to be beyond the capabilities of my computer.

Is there some method to solve something like [A,B]=C, where B is the unknown operator?

Basically I want to solve the equation

[D,A]=iaL

_{z,}where D is the Laplacian (in cylindrical coordinates), a is some constant, and A is the unkown operator.

I know this question has answer for when G is infinite cyclic group group.Does there is a general proof?

Hi all

I add here a pdf file about my question. The question involves an "easier" parametrization which I found and is described here. This is not a matter of life and

death, but if somebody can think something easier I would be very grateful.

I can live with the parametrization that I found, but what is bugging me, is the fact that

I might be missing some much easier and obvious parametrization...

kind regards

Samuli Piipponen

Type means the vector space dim of the first non zero Ext w.r.t R/m

I know the Milnor and Tjurina numbers reflect in certain way the complexity of

a singular point for a variety defined by one polynomial V=v(<f(x_1,...,x_n>)).

Now I know there exists some generalizations (or at least attempts have been made to generalize) similar quantities for a variety defined by more than one polynomial V(f_1,...,f_n).

For example the Milnor number should generalize to object called 'Milnor Class' in Chow's group.

I have mainly 2 questions regarding these possible generalizations:

1) Does anybody know a good reference book in which such problems would have been investigated, discussed and possibly rigorously defined.

2) Are there any known algorithms to compute such objects ? And if so does anyone know an article or a book where such algorithms could have been described ? Or at least partial results ?

Is there a connection to computer science (theoretical), statistics, etc? I know algebraic statistics is a field where they study graphical models by viewing it as algebraic varieties, but I don't know much. Does anyone know of any applications?

See the following publication: Yahya Talebi and M. J. Nematollahi, "Modules with C*-condition", Taiwanese J. Math. (2009)

Consider when n is not a prime number, how many annihilator ideals are there when there is a factorization like this?