User miguel - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-19T13:18:27Z http://mathoverflow.net/feeds/user/20272 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/114272/chain-condition-on-rings Chain Condition on Rings Miguel 2012-11-23T18:14:12Z 2012-11-23T18:50:56Z <p>Let $R$ be a noncommutative ring. The ring $R$ has descending chain condition on two-sided ideals (D.C.C.), if for a chain of two-sided ideals $J_1\supset J_2\supset \cdots$, then there exists an $N\in\mathbb N$ such that $J_n = J_N$ for $n\geq N$. </p> <p>Is there any example of a ring $R$ with D.C.C. on two-sided ideals but without D.C.C on left ideals and also without D.C.C on right ideals? </p> http://mathoverflow.net/questions/109339/commuting-linear-operators-in-hilbert-spaces Commuting Linear Operators In Hilbert Spaces Miguel 2012-10-10T23:05:28Z 2012-10-12T04:01:29Z <p>Let $V$ be a finite dimensional vector space over the complex field $\mathbb C$. Let $L:V\rightarrow V$ be a linear operator. Using the matrix of $L$ and the Jordan canonical form it is easy to find all the linear operators that commute with $L$. </p> <p>Now suppose that $H$ is a Hilbert space and let $L:H\rightarrow H$ be a continuous linear operator. There is some method to determine all the continuous linear operatores that commute with $L$?</p> http://mathoverflow.net/questions/109234/inner-automorphisms-of-matrix-algebras Inner Automorphisms of Matrix Algebras Miguel 2012-10-09T14:55:21Z 2012-10-09T18:23:13Z <p>Let $\mathbb R$ be the field of real numbers and $\mathbb C$ the field of complex numbers. It is well known that that $\mathbb C$ can be embedded in $M_2(\mathbb R)$.</p> <p>This embedding can be extended in the obvious way to an embedding function $\varphi : M_n(\mathbb C)\rightarrow M_{2n}(\mathbb R)$. </p> <p>My question is: consider the embedding $\varphi :M_2(\mathbb C)\rightarrow M_{4}(\mathbb R)$, it is possible to find an inner automorphism $\Psi : M_4(\mathbb R)\rightarrow M_{4}(\mathbb R)$ such that the intersection $$\varphi (M_2(\mathbb C))\cap \Psi(\varphi (M_2(\mathbb C)))$$</p> <p>is the scalar matrices in $M_4(\mathbb R)$?</p> http://mathoverflow.net/questions/108402/decomposition-of-matrices-in-semisimple-and-nilpotent-parts Decomposition of Matrices in Semisimple and Nilpotent Parts Miguel 2012-09-29T11:56:20Z 2012-09-30T15:38:58Z <p>I asked this question in <a href="http://math.stackexchange.com/posts/204115/edit" rel="nofollow">http://math.stackexchange.com/posts/204115/edit</a> ​​but remains unanswered.</p> <p>For any matrix $A\in M_n(\mathbb F)$, where $\mathbb F$ is an algebraically closed field, there is a matrix $S\in M_n(\mathbb F)$ such that </p> <p>$$SAS^{-1}=D+N,$$ where $D$ is diagonal and $N$ nilpotent. Moreover, this decomposition is unique.</p> <p>Suppose now that $A\in M_n(\mathbb K)$, but $\mathbb K$ is not necessarily algebraically closed. It is also true that there is a matrix $L\in M_n(\mathbb K)$ such that </p> <p>$$LAL^{-1}=R+M,$$</p> <p>where $M$ is nilpotent and $R$ is diagonalizable in the algebraic closure of $\mathbb K$? Moreover when we consider the decomposition in $\mathbb K$ and in the algebraic closure of $\mathbb K$ the nilpotent part is the same? </p> http://mathoverflow.net/questions/99446/annihilators-in-algebras Annihilators in algebras Miguel 2012-06-13T13:45:47Z 2012-06-13T16:14:41Z <p>Let $R$ be an artinian commutative ring and $A$ an algebra over $R$ with basis ${a_1, a_2, \ldots , a_n}$ where each $a_i$ is a unit and</p> <p>$a_ia_j=u_{ij}a_ja_i,$</p> <p>where each $u_{ij}\in R$ and is a unit. is true that for every element $x\in A$ the left annihilator of $x$ denoted by $ann_l{x}$ equals the right annhilator $ann_r{x}$?</p> <p>I think that if exist an element $z$ with $ann_l{z}\neq ann_r{z}$, this lack of symmetry must necessarily be reflected in the ring $R$ or in the elements of the base. Since, for the ring $R$ and for the elements of the base there is symmetry, this symmetry extends to all elements of $A$. This is my perception, but how to prove? </p> http://mathoverflow.net/questions/98883/finite-local-rings Finite local rings Miguel 2012-06-05T15:38:04Z 2012-06-06T07:20:09Z <p>There is some classification of finite commutative local rings. For example how many not isomorphic finite local rings with the same order $p^k$ and the same residue field $\mathbb F_p$ exist?</p> http://mathoverflow.net/questions/98272/annihilators-in-artinian-ring Annihilators in artinian ring Miguel 2012-05-29T12:44:01Z 2012-05-29T13:18:26Z <p>Let $A$ be an artinian ring. It is well known that for an element $x$ in $R$ the right annihilator $Ann_r(x)$ is non trivial (i.e. contains a nonzero element ) if and only if the left annihilator $Ann_l(x)$ is non trivial. </p> <p>If we assume that the ring $A$ is finite is true that the cardinality of $Ann_r(x)$ equals the cardinality of $Ann_l(x)$?</p> <p>Note: If the ring $A$ is finite and semisimple, them it is a direct sum of full matrix rings over fields and in this case it is true. I think it may help to think first in full matrix rings over finite commutative rings. </p> http://mathoverflow.net/questions/90825/permanent-function-over-finite-commutative-rings Permanent function over finite commutative rings Miguel 2012-03-10T15:20:26Z 2012-03-10T15:20:26Z <p>Let $R$ be a finite commutative ring with identity and $R=R_1+...+R_k$ its decomposition in a direct sum of finite local rings. Considere the matrix ring $M_n(R)$ of $n\times n$ matrices with elements in $R$. It is easy to see that a matrix $A$ decomposes in a unique way as </p> <p>$$A=A_1+ .... +A_k,$$</p> <p>with $A_i\in R_i$. Let $per(A)$ denote the permanent of the matrix $A$.</p> <p>Is true that $per(A)$ is a unit in $R$ if and only if each $per(A_i)$ is a unit in $A_i$?</p> http://mathoverflow.net/questions/90157/determinants-over-commutative-rings Determinants over commutative rings Miguel 2012-03-03T22:42:22Z 2012-03-04T02:36:53Z <p>Hello, I am preparing a paper on determinants in commutative rings. Someone can give me examples of applications of determinants in commutative rings to other areas of mathematics or physics. Thank you</p> http://mathoverflow.net/questions/88930/online-free-copy-of-the-article-what-is-an-answer-amer-math-monthly/88934#88934 Answer by Miguel for Online free copy of the article "What is an answer?" (Amer. Math. Monthly) Miguel 2012-02-19T12:57:59Z 2012-02-19T12:57:59Z <p>Hello. send me your email and I send you a copy of the article "What is an Answer? Amer. Math. Monthly, 89 (5), pp. 289-292"</p> http://mathoverflow.net/questions/86217/tensor-product-of-two-algebras Tensor product of two algebras Miguel 2012-01-20T15:26:24Z 2012-01-20T15:26:24Z <p>Let $A$ and $B$ be algebras over a field $K$. The ideals of the tensor product $A\bigotimes_K B$ are of the form $I\bigotimes_K J$ where $I$ is an ideal of $A$ and $J$ is an ideal of $B$?</p> http://mathoverflow.net/questions/85915/maximal-ideals-in-a-polynomial-ring-over-the-real-numbers Maximal ideals in a polynomial ring over the real numbers. Miguel 2012-01-17T16:23:30Z 2012-01-17T18:43:09Z <p>Let $\mathbf{R}$ be the field of real numbers. What are the generators of the maximal ideals of the polynomial ring $\mathbf{R}[x_1, ... , x_n]$? If instead of $\mathbf{R}$ one considers the field $\mathbf{C}$ of complex numbers, then Hilbert's Nullstellensatz implies that each maximal ideal $\mathfrak{m}$ of $\mathbf{C}[x_1, ... , x_n]$ is generated by $n$ generators of the form $x_i-a_i$. </p> http://mathoverflow.net/questions/85836/noetherian-ring Noetherian ring Miguel 2012-01-16T19:25:16Z 2012-01-17T01:06:47Z <p>Let $R$ be a noetherian ring. By the Hilbert Basis Theorem the polinomial ring $R[x_1, ... , x_n]$ is also a noetherian ring. What can we say about the number of generators of an ideal $I$ of $R[x_1, ... , x_n]$? (We can suppose that every ideal in $R$ is principal)</p> http://mathoverflow.net/questions/84591/subalgebra-of-a-matrix-algebra subalgebra of a matrix algebra Miguel 2011-12-30T14:33:29Z 2011-12-30T21:18:56Z <p>Let $K$ be an algebraic closed field and $M_n(K)$ the $K$-algebra of all matrices $n\times n$ over $K$. If $L$ and $M$ are two commutative isomorphic subalgebras of $M_n(K)$ it is true that there exista a regular matrix $S\in M_n(K)$ such that $SLS^{-1}=M$. That is the isomorphism is inner? </p> http://mathoverflow.net/questions/114272/chain-condition-on-rings/114276#114276 Comment by Miguel Miguel 2012-11-23T18:46:54Z 2012-11-23T18:46:54Z Muchas gracias Fernado. http://mathoverflow.net/questions/99446/annihilators-in-algebras/99468#99468 Comment by Miguel Miguel 2012-06-13T18:35:07Z 2012-06-13T18:35:07Z Thank you. It is a nice example to prove that we have not $ann_l x=ann_rs$. But, probably, their cardinality is the same. I'm trying to find a proof of this. http://mathoverflow.net/questions/98883/finite-local-rings Comment by Miguel Miguel 2012-06-06T09:41:39Z 2012-06-06T09:41:39Z Martin, I do not understand your comment, you want to explain? http://mathoverflow.net/questions/98883/finite-local-rings/98931#98931 Comment by Miguel Miguel 2012-06-06T07:37:56Z 2012-06-06T07:37:56Z Thanks, your answer is very useful for me. http://mathoverflow.net/questions/98883/finite-local-rings Comment by Miguel Miguel 2012-06-05T17:10:29Z 2012-06-05T17:10:29Z Gracias Mariano http://mathoverflow.net/questions/98272/annihilators-in-artinian-ring/98275#98275 Comment by Miguel Miguel 2012-05-29T17:05:06Z 2012-05-29T17:05:06Z thanks, is a very clever example. http://mathoverflow.net/questions/90157/determinants-over-commutative-rings Comment by Miguel Miguel 2012-03-04T00:17:23Z 2012-03-04T00:17:23Z The paper is on the convertion the determinant into the permanent. Yes i want applications where the ring has zero divisors. http://mathoverflow.net/questions/85915/maximal-ideals-in-a-polynomial-ring-over-the-real-numbers/85916#85916 Comment by Miguel Miguel 2012-01-17T17:47:05Z 2012-01-17T17:47:05Z Thank you Francesco Polizzi http://mathoverflow.net/questions/84591/subalgebra-of-a-matrix-algebra/84600#84600 Comment by Miguel Miguel 2011-12-30T18:05:13Z 2011-12-30T18:05:13Z thank you Denis Serre. Your example responds to my question. Your answer made ​​me think of another question: two unital subalgebras (that is, containing the scalar matrices) are conjugated? http://mathoverflow.net/questions/84591/subalgebra-of-a-matrix-algebra Comment by Miguel Miguel 2011-12-30T15:23:12Z 2011-12-30T15:23:12Z Dear darij grinberg thanks for your suggestion. But I think it can not be applied directly the Skolem–Noether theorem since the subalgebra $A$ is not necessarily simple. -Miguel