User zacarias - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T01:53:09Z http://mathoverflow.net/feeds/user/24864 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105040/centralizer-of-a-matrix-over-a-finite-field Centralizer of a Matrix over a Finite Field zacarias 2012-08-19T15:36:31Z 2013-05-24T18:04:30Z <p>This question in stackExchange remained unanswered. </p> <p>Let $\mathbb F$ be a finite field. Denote by $M_n(\mathbb F)$ the set of matrices of order $n$ over $\mathbb F$ . For a matrix $A∈M_n(\mathbb F)$ what is the cardinality of $C_{M_n(\mathbb F)} (A)$ , the centralizer of $A$ in $M_n(\mathbb F)$? There are papers about it? </p> http://mathoverflow.net/questions/115310/matrices-over-finite-prime-fields Matrices over Finite Prime Fields zacarias 2012-12-03T16:24:08Z 2012-12-03T17:03:53Z <p>Let $p$ be an odd prime and $\mathbb Z_p$ be the prime field of order $p$. Consider the matrix ring $R=M_n(\mathbb Z_p)$. Is there any method to count the solutions of the equation (in the ring $R$)</p> <p>$$X^2=I.$$ Where $I$ is the identity matrix?</p> http://mathoverflow.net/questions/110702/factor-rings-of-the-ring-of-integers-in-a-number-field Factor Rings of the Ring of integers in a Number field zacarias 2012-10-25T21:18:09Z 2012-10-26T07:33:41Z <p>The factor rings of the ordinary integers $\mathbb Z$ are the well-known residual classes $\mathbb Z_n$. For the Gaussian integers $\mathbb Z[i]$ the factor rings are studied in </p> <p>1) J. T. Cross, The Euler ϕ-function in the Gaussian integers, Amer. Math. Monthly 90 (1983) 518–528.</p> <p>2) Dresden, Greg(1-WLEE); Dymàček, Wayne M.(1-WLEE) Finding factors of factor rings over the Gaussian integers. Amer. Math. Monthly 112 (2005), no. 7, 602–611. </p> <p>There are articles on the factor rings of the ring of integers of an algebraic number field?</p> http://mathoverflow.net/questions/104196/commutative-subrings-of-finite-matrix-rings Commutative Subrings of Finite Matrix Rings zacarias 2012-08-07T13:05:10Z 2012-08-07T18:47:09Z <p>Let $R$ be a finite commutative ring with identity. Considere the matrix ring $A=M_n(R)$. What is the order of a maximal commutative subring of $A$ that contains all scalar matrices?</p> http://mathoverflow.net/questions/103450/embedding-semigroups-in-rings Embedding Semigroups in Rings zacarias 2012-07-29T14:09:57Z 2012-07-30T15:59:26Z <p>Let $S$ be a finite commutative semigroup with identity. Under what conditions (on the semigroup $S$) it is possible to find a ring $R$ such that the multiplicative structure of $R - \{0\}$ is isomorphic to $S$? </p> http://mathoverflow.net/questions/103298/matrix-algebras Matrix Algebras zacarias 2012-07-27T11:58:55Z 2012-07-27T13:04:23Z <p>I'm reading the papaer "On the Reduction of a Matrix to Diagonal Form" of Epstein and Flanders (Amer. Math. Monthly 62, (1955). 168–171.</p> <p>Let $S$ denote the trace function.</p> <p>The authors stated that a well-known result in the theory of algebras of matrices is: </p> <p>A matrix algebra $\mathbb U$ over a field $\mathbb F$ of characteristice zero is semisimples if and only if $S(XY)=0$, for fixed $X\in\mathbb U$ and all $Y\in\mathbb U$,implies $X=0$.</p> <p>The authors do not give references. Does anyone know where I can find a proof of this result?</p> http://mathoverflow.net/questions/102164/annihilators-in-matrix-rings Annihilators in Matrix Rings zacarias 2012-07-13T18:40:19Z 2012-07-13T22:33:56Z <p>I think this is not a research question, but in stackExchange remained unanswered.</p> <p>Let $R$ be a finite commutative ring. For $n>1$ consider the full matrix ring $M_n(R)$ . For a matrix $A\in M_n(R)$ is true that the cardinality of the left annihilator (in $M_n(R)$ ) of $A$ equals the cardinality of the right annhilator?</p> http://mathoverflow.net/questions/101674/proof-a-la-grothendieck Proof " A la Grothendieck" zacarias 2012-07-08T14:49:19Z 2012-07-08T14:49:19Z <p>Grothendieck's proof of the well-known Ax–Grothendieck Theorem uses the elegant argument that the existence of a counterexample for the field $\mathbb C$ imply the existence of a counterexample for a finite field. Roughly speaking, we transfer the problem from $\mathbb C$ to finite fields.</p> <p>recently, Will Sawin, here in mathoverflow, proved that the ring of Hamilton quaternions over $\mathbb H_{2^s}$ is reversible proving that the existence of a counterexample fore some $s$ imply the existence of a counterexample for $s=1$ or $s=2$, (a finite problem tha is solved with a computer). </p> <p>There are other examples of this type of proof? </p> http://mathoverflow.net/questions/101230/probability-in-the-primes Probability in the Primes zacarias 2012-07-03T14:00:34Z 2012-07-03T15:58:12Z <p>Given two randomly chosen positive rational integers, the probability that the two numbers are coprime is $\frac{6}{\pi^2}$. This is also the probability that a positive integer is squarefree. Are there generalizations of these results for Gaussian integers? Or more generally for the ring of integers in an algebraic number field?</p> http://mathoverflow.net/questions/105040/centralizer-of-a-matrix-over-a-finite-field/105076#105076 Comment by zacarias zacarias 2012-08-22T15:40:40Z 2012-08-22T15:40:40Z Thanks Amritanshu http://mathoverflow.net/questions/105040/centralizer-of-a-matrix-over-a-finite-field/105042#105042 Comment by zacarias zacarias 2012-08-22T15:40:19Z 2012-08-22T15:40:19Z Thanks Geoff Robinson http://mathoverflow.net/questions/105040/centralizer-of-a-matrix-over-a-finite-field/105116#105116 Comment by zacarias zacarias 2012-08-22T15:39:41Z 2012-08-22T15:39:41Z Thanks Alexander. http://mathoverflow.net/questions/105040/centralizer-of-a-matrix-over-a-finite-field/105052#105052 Comment by zacarias zacarias 2012-08-19T20:49:57Z 2012-08-19T20:49:57Z Thanks Alireza Abdollahi http://mathoverflow.net/questions/105040/centralizer-of-a-matrix-over-a-finite-field Comment by zacarias zacarias 2012-08-19T16:56:18Z 2012-08-19T16:56:18Z <a href="http://math.stackexchange.com/questions/184275/centralizer-of-a-matrix-over-a-finite-field" rel="nofollow" title="centralizer of a matrix over a finite field">math.stackexchange.com/questions/184275/&hellip;</a> http://mathoverflow.net/questions/104196/commutative-subrings-of-finite-matrix-rings Comment by zacarias zacarias 2012-08-07T15:56:50Z 2012-08-07T15:56:50Z Yes, the interesting problem is for finite rings. http://mathoverflow.net/questions/104196/commutative-subrings-of-finite-matrix-rings Comment by zacarias zacarias 2012-08-07T15:30:03Z 2012-08-07T15:30:03Z well, we must impose restrictions on the ring $R$. If $R$ is finite maximal subrings mean subring of maximal order. If $R$ is infinite suppose that $R$ is a principal ideal domain. http://mathoverflow.net/questions/104196/commutative-subrings-of-finite-matrix-rings Comment by zacarias zacarias 2012-08-07T13:44:45Z 2012-08-07T13:44:45Z I think that a non-free subalgebra is never maximal, because is contained in a free algebra. For example in the example of Florian Eisele if we take $b\in R$ (and not in the ideal I) the subring is free. http://mathoverflow.net/questions/104196/commutative-subrings-of-finite-matrix-rings Comment by zacarias zacarias 2012-08-07T13:39:41Z 2012-08-07T13:39:41Z Yes, I mean free as an $R$- module. http://mathoverflow.net/questions/103450/embedding-semigroups-in-rings/103520#103520 Comment by zacarias zacarias 2012-07-30T22:33:29Z 2012-07-30T22:33:29Z Thank you Mark Sapir http://mathoverflow.net/questions/103450/embedding-semigroups-in-rings Comment by zacarias zacarias 2012-07-29T16:04:56Z 2012-07-29T16:04:56Z thank you Benjamin http://mathoverflow.net/questions/103298/matrix-algebras/103304#103304 Comment by zacarias zacarias 2012-07-27T13:09:29Z 2012-07-27T13:09:29Z thank you Bugs Bunny http://mathoverflow.net/questions/102164/annihilators-in-matrix-rings/102189#102189 Comment by zacarias zacarias 2012-07-14T13:47:57Z 2012-07-14T13:47:57Z Thank you Konstantin. This is a nice counterexample. I think that the answer is yes in the case in which the finite commutative ring has identity (the proof of Ralph assumes that the ring has identity ). http://mathoverflow.net/questions/102164/annihilators-in-matrix-rings/102176#102176 Comment by zacarias zacarias 2012-07-13T20:28:26Z 2012-07-13T20:28:26Z Thanks for the reply David. But my question is for a commutative ring $R$. In your answer $x$ and $y$ don't commute and therefore the ring $R$ is noncommutative. http://mathoverflow.net/questions/101539/linear-systems-over-commutative-rings Comment by zacarias zacarias 2012-07-06T23:39:12Z 2012-07-06T23:39:12Z Thank you dear Qiaochu and dear Andreas. If fact it is not a research-level question, so I also vote to close.