Tagged Questions

4answers
112 views

1answer
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What would be an example such that $aH=bH$ but $Ha \neq Hb$?

Let $H$ be a subgroup of a group $G$. Assume $aH=bH$ for some $a,b\in G$ What would be an example such that $Hb\neq Ha$? I cannot imagine what would be.. Moreover, if $H$ is infinite, ...
1answer
66 views

Inner Product on a Vector Space over a field besides $\mathbb R$ or $\mathbb C$?

Are there any fields with vector spaces you can define an inner product over besides subfields of $\mathbb C$? I know that you'd want the field to contain an ordered subfield, so it must have ...
2answers
329 views

If the tensor power $M^{\otimes n} = 0$, is it possible that $M^{\otimes n-1}$ is nonzero?

Let $M$ be a module over a commutative ring $R$. It is possible that $M \otimes M = 0$ if $M$ is nonzero, for example when $R = \mathbb{Z}$ and $M = \mathbb{Q}/ \mathbb{Z}$. What about when higher ...
1answer
59 views

A question about polynomials in $K[x_1,x_2,…,x_n]$ and there permutations

Let $K$ be a field. Let $n$ be a positive integer and $P$ be a non-symmetric polynomial in $K[x_1,x_2,...,x_n]$. $S_n$ acts on $K[x_1,...,x_n]$ in an obvious way. Let $P_1,P_2,...,P_r$ be the ...
0answers
52 views

Heineken and Mohamed Groups

I was trying to understand the costruction of the Heineken and Mohamed groups. (II) Every proper subgroup of $G$ is subnormal and nilpotent. Lemma 1. If $G$ is a non-nilpotent soluble group ...
2answers
102 views

Can we make $\mathbb{Z}$ into a field?

This is probably an elementary question about fields, but I think it is a little tricky. Can we make the integers $\mathbb{Z}$ into a field? Let me be more precise. Is it possible to make ...
1answer
86 views

Localization of an additive category which is no longer additive

Is there a nice example of an additive category $C$ and a family of morphisms $S\subset Mor(C)$ such that $C[S^{-1}]$ is no longer additive? I know that in general localization of categories ...
1answer
82 views

Weakened associativity axiom: $(x * y) * z \leq x*(y*z).$

Call a partially ordered magma $(X,*)$ sub-associative iff it satisfies the following axiom. $$(x*y)*z \leq x*(y*z).$$ Basically, this is saying that we may shuffle brackets right to get a larger ...
3answers
167 views

Counterexamples to Nakayama's Lemma if $M$ is not finitely generated

One of the most famous forms of Nakayama's lemma says: Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module. If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, ...
3answers
189 views

Counterexample to linear transformation.

What counterexample can I use to prove that ($\mathbb{R}_{[x]}$is any polynomial): $L :\mathbb{R}_{[x]}\rightarrow\mathbb{R}_{[x]},(L(p))(x)=p(x)p'(x)$ is not linear transformation. I have already ...
1answer
83 views

If all the centralizers are finite, then does it follow that the group is finite?

As far as I have heard, centralizers play an important role in the theory of groups. My question arises from curiosity and the desire to understand how much control centralizers have over the group. ...
3answers
159 views

What are some examples of “exotic” algebraic structures? [closed]

I guess that I'm quite familiar with the basic "everyday algebraic structures" such as groups, rings, modules and algebras and Lie algebras. Of course, I also heard of magmas, semi-groups and monoids, ...
2answers
104 views

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$? [closed]

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$ ? Im sorry I am confused about ring theory. Its all new to me.
1answer
214 views

Examples of non-isomorphic fields with isomorphic group of units and additive group structure

YACP mentions in a comment that: There are examples of non-isomorphic fields $K$ and $L$ with $(K,+)\cong (L,+)$ and $(K^{\times} ,\cdot)\cong (L^{\times},\cdot)$ Can someone provide an ...
2answers
158 views

Counterexample: multiplying modules by elements of an ideal vs. taking linear combinations

Let $R$ be a ring (commutative, unital) and $M$ an $R$-module. Let $I \subset R$ be an ideal. We make the following definitions: $$A := \{ am \ | \ a \in I,\ m \in M \}$$  B := \left\{ ...
2answers
75 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
4answers
1k views

Example of a finite non-commutative ring without a unity

Give an example of a finite, non-commutative ring, which does not have a unity. I can't think of any thing which fits this question. I was thinking $M_2(\mathbb{R})$ but it has the identity. Any ...
3answers
84 views

Examples of root, parabolic, and borel subgroups corresponding to roots

I'm interested in seeing a few examples of root, parabolic, and Borel subgroups given a specific reductive group $G$. Here is what I know. Let $G$ be a reductive algebraic group over an ...
2answers
117 views

Do silly-rings exist?

A ring can be defined as a near-ring satisfying two-sided distributivity, whose underlying additive group is Abelian. Negating this second stipulation, we obtain the following definition. A silly-ring ...
3answers
105 views

Structures with addition, multiplication and exponentiation.

The set $\mathbb{N}$ can be viewed as a mathematical structure with operations off addition, multiplication and exponentiation. Observe that: It forms an Abelian monoid under both addition and ...
6answers
381 views

If $G$ is a group and $N$ is a nontrivial normal subgroup, can $G/N \cong G$? [duplicate]

I know $G/N$ is isomorphic to a proper subgroup of $G$ in this case, so the gut instinct I had was 'no'. But there are examples of groups that are isomorphic to proper subgroups, such as the integers ...
1answer
39 views

Irreducibility of Lie algebra representations

Let $\mathfrak{g}$ be a finite dimensional real Lie algebra and $\pi: \mathfrak{g} \to \mathfrak{gl}(V)$ be a homomorphism of real Lie algebras where $V$ is a finite dimensional real vector space. But ...
4answers
137 views

Number of elements in a group and its subgroups (GS 2013)

Every countable group has only countably many distinct subgroups. The above statement is false. How to show it? One counterexample may be sufficient, but I am blind to find it out. I have ...
2answers
57 views

A non-UFD where we have different lengths of irreducible factorizations?

A unique factorization domain (UFD) is defined to be an integral domain where each element other than units and zero has a factorization into irreducibles, and if we have two such factorizations then ...
3answers
110 views

Can an object be universal with respect to several properties?

Does anyone know an example of an object $A$ in some category $\mathcal C$ such that $A$ satisfies at the same time more than one universal property? Of course, when I say that $A$ is universal with ...
1answer
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1answer
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Objects with a “Homogeneity Principle”

So I don't have to worry about formalities, in the following let $\mathscr{C}$ be a sufficiently nice category--at least nice enough so that the following definition makes sense. I believe concrete ...
1answer
261 views

Extending Herstein's Challenging Exercise to Modules

Anybody who has worked through Herstein's Topics in Algebra might remember Exercise 26 of Section 2.5 (in second edition): If $G$ is an abelian group containing subgroups of order $m$ and $n$, ...
1answer
82 views

Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
2answers
126 views

Must certain rings be isomorphic to $\mathbb{Z}[\sqrt{a}]$ for some $a$

Consider the group $(\mathbb{Z}\times\mathbb{Z},+)$, where $(a,b)+(c,d)=(a+c,b+d)$. Let $\times$ be any binary operation on $\mathbb{Z}\times\mathbb{Z}$ such that ...
4answers
133 views

Infinity and structures

Do you know any case (example) where an "infinite" object with a structure (say, an infinite group) cannot be extended (in the sense of adding elements) in any way without it no longer having the ...
4answers
158 views

Example of a module for non-mathematicians

I'm looking for a non-trivial1 example of a module that would be recognizable to a non-mathematician. I.e. I'm looking for examples of modules that one may come across in "the real world". The ...
3answers
105 views

Ring and Subring with different Identities [duplicate]

Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know ...