All questions with tag [math: abstract-algebra]


Algebra without Zorn is lemma

One can not get also much in abstract algebra prior to running into Zorn is Lemma. As an example, it is made use of in the evidence that every nonzero ring has a topmost perfect. Nonetheless, it appears that if we limit our emphasis to Noetherian rings, we can usually stay clear of Zorn is lemma. Just how much could a growth of the concept for s...
2019-12-06 14:37:43

Rings of matrices

Let $ A\in {\mathbb{F} }^{n\times n} $ be a dealt with matrix. The set of all matrices that commute with A kinds a subring of ${\mathbb{F} }^{n\times n}$. Is any kind of subring of ${\mathbb{F} }^{n\times n }$ (which has the identification) of the above kind? Many thanks.
2019-12-06 14:37:16

What is it called when a subalgebra has its centralizer?

In the inquiry Math.SE #16716, Natalia inquired about standing for rings of matrices as centralizers of a matrix. This is a fascinating inquiry, yet had some clear troubles as rings of matrices require not have their very own centralizers, which would certainly be rather negative. In limited teams, subgroups which contain their very own central...
2019-12-06 14:34:43

Is a monoid finitely created by routine matrices routine?

This inquiry is rather connected to this one - - actually, I suggested to ask the adhering to : Let $\mathcal{M}_n$ be the multiplicative monoid of $n \times n$ matrices on $\mathbb{N}$ (consisting of $0$). Allow $M_1, \ldots, M_k \in \mathcal{M}_n$ and also intend the monoid created by every one of them is limited. Is the monoid created by al...
2019-12-06 14:31:22

Equations over $\mathbb Z/p^e \mathbb Z$

Let $\{f_i\}$ be a system of straight formulas in $X_1,...,X_n$ with coefficients in $R = \mathbb Z/p^e \mathbb Z$ (i.e. modulo a primepower). Think there is an one-of-a-kind remedy $a_i$. $f_1 = u g_1$ where $u$ is a non - device, i.e. $p\mid u$. Is it real that $\{f_i\mid i>1\}$ allows just the very same one-of-a-kind remedy? Or ...
2019-12-06 14:29:52

Does the formula $x^4+y^4+1 = z^2$ have a non - unimportant remedy?

The history of this inquiry is this : Fermat confirmed that the formula, $$x^4+y^4 = z^2$$ has no remedy in the favorable integers. If we take into consideration the close to - miss out on, $$x^4+y^4-1 = z^2$$ after that this has lots (actually, an infinity, as it can be addressed by a Pell formula). Yet J. Cullen, by extensive search, located...
2019-12-06 14:22:09

Historical book on team theory/algebra

Recently I have actually begun reviewing several of the background of maths in order to much better recognize points. A great deal of suggestions in algebra originated from attempting to recognize the trouble of locating remedies to polynomials in regards to radicals, which is addressed by the Abel - Ruffini theory and also Galois concept. I w...
2019-12-06 14:20:53

Unique factorization domain name that is not a Principal excellent domain name

Let $c$ be an integer, not always favorable and also not a square. Allow $R=\mathbb{Z}[\sqrt{c}]$ represent the set of varieties of the kind $$a+b\sqrt{c}, a,b \in \mathbb{Z}.$$ Then $R$ is a subring of $\mathbb{C}$ under the common enhancement and also reproduction. My inquiry is : if $R$ is a UFD (one-of-a-kind factorization domain name), doe...
2019-12-06 14:19:29

Count of components in $\Bbb{Z}_7[x]/(3x^2+2x)$

Hi I have some trouble just how to get matter of components in $\Bbb{Z}_7[x]/(3x^2+2x)$. I assume there come from just polynomials which are indivisible with $3x^2+2x$ ($\gcd=1$). I assume it is so as much I recognize that as an example in every $\Bbb{Z}_m$, $m$ prime, is the matter of belonging components eqauls to $\phi(m)$. Yet I actually do ...
2019-12-06 14:18:41

Division of Other contours than circles

The works with of an arc of a circle of size $\frac{2pi}{p}$ are an algebraic number, and also when $p$ is a Fermat prime you can locate it in regards to square origins. Gauss claimed that the method related to a whole lot extra contours than the circle. Will you please inform if you recognize any kind of functioned instances of this (locating ...
2019-12-06 14:14:34

Why is this appropriate mostly injective ring a self - injective ring?

If $R$ is a semiprime ring, appropriate mostly injective and also pleases ACC on appropriate annihilators of components, is it self - injective? I just recognize that it is appropriate nonsingular.
2019-12-06 14:10:17

Similarity in between reduction and also department

I would love to listen to some intuition concerning distinction in between reduction and also department. For binary reduction driver the typical growth is intro of unary procedure of taking adverse number. For department nonetheless it relies on the domain name. For "nice" domain names (such as sensible and also intricate numbers) we ...
2019-12-06 13:52:33

Applications of cubic in number theory?

The remedy of the cubic formula is recognized in regards to a sensible function of a dice origin of a square origin. If we simply need to know the value it is very easy to approximate it making use of a mathematical method. I would love to recognize if the real kind (in regards to origins) of the remedy of the cubic has any kind of applications ...
2019-12-06 13:39:28

Epimorphism from GL (2, Z) to GL (2, Z)

Is there an epimorphism $f\colon \mathrm{GL}(2,\mathbb{Z})\to \mathrm{GL}(2,\mathbb{Z})$ which is not injective? Below, $\mathrm{GL}(2,\mathbb{Z})$ is the team of invertible $2\times 2$ matrices with integer access. Included. The solution is no : this team is Hopfian. I recognize an evidence in which it is revealed that this team is finitely ...
2019-12-05 03:24:26

Sylow Subgroup trouble

I was asked to confirm : no team can have a marginal regular subgroup isomorphic to a $\mathrm{Syl}_2(A_7)$. I assume I need to locate some building that $\mathrm{Syl}_2(A_7)$ has yet not a marginal regular subgroup. So after that I assumed that it a $\mathrm{Syl}_2(A_7)$ constantly has particular subgroup, i.e. exist $G\ char\ \mathrm{Syl}_2(A...
2019-12-04 01:38:15

Is there an indispensable domain name with a great deal of deposit areas of the very same feature?

Exists a commutative indispensable domain name $R$ in which : every nonzero prime excellent $Q$ is topmost, and also for every single prime power $q\equiv 3 \bmod 8$, there is a topmost excellent $Q$ of $R$ such that $R/Q$ is an area of dimension $q$? I am seeking a ring where "$q\equiv 3 \bmod 8$" defines limited areas, as op...
2019-12-04 01:36:23

Does the order, latticework of subgroups, and also latticework of variable teams, distinctly establish a team approximately isomorphism?

If we have a 2 latticeworks (partly gotten) - one for subgroups, one for variable teams, and also we understand order of the team we intend to have these subgroup and also variable team latticeworks, is such a team one-of-a-kind approximately isomorphism (if exists)? Or exists a counterexample? If that holds true, suffice problems on the order ...
2019-12-04 01:09:45

Can the tensor item of 2 non - free abelian groups be non - absolutely no free?

It is rather very easy to construct an (R - S) bi - component M and also a left S - component N such that neither M_S neither N is a projective S - component, yet the tensor item is a non - absolutely no projective R - component. Nonetheless, taking R = S = Z opposes my zoo of instances. Below projective = free is shut under straight amounts an...
2019-12-04 01:02:14

Count of one-of-a-kind algebras with one unary procedure (on limited set)

It is feasible, that matter of one-of-a-kind algebras (approximately isomorphism) with one unary procedure on set with $n$ components is $2^n-1$? For $n=1,2,3,4$ is this theory real (I still have actually not validated it on the computer system). Exists any kind of counterexample, or suggestion for evidence? Many thanks.
2019-12-04 00:59:30

Cardinality of Two Sets of Points

The trouble is mentioned - Do the adhering to 2 collections of factors have the very same cardinality and also if so, develop a bijection : A line sector of size 4 and also fifty percent of the area of distance one (consisting of both endpoints). My thinking is they do have the very same cardinality and also my bijection is an image in which ...
2019-12-04 00:44:39