0
votes
2answers
26 views

Find the number of prime ideals?(CSIR 2014)

Let $p,q$ be distinct primes. Then (1) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct ideals. (2) $\dfrac{\mathbb{Z}}{p^2q}$ has exactly 3 distinct prime ideals. (3) $\dfrac{\mathbb{Z}}{p^2q}$ ...
2
votes
1answer
50 views

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite.

1) Show that $(\Bbb Z[\sqrt2]^*, .)$ is infinite. 2) Classify $(\Bbb Z[\sqrt2]^*, .)$, where $\Bbb Z[\sqrt2]^*$ is the group of units of $\Bbb Z[\sqrt2]$ What I have done so far that for $a+b\sqrt2$ ...
0
votes
0answers
48 views

Realize groups as unit group of a ring

Let $A$ be a ring, $G$ be a group, and $f:A^{\times} \rightarrow G$ be a group homomorphism. Is there any ring $B$ and ring homomorphism $\varphi:A \rightarrow B$ such that $G$ is subgroup of ...
2
votes
0answers
27 views

prove that $(E_{p^n},*)$ is cyclic group

if $p \in$ $\mathbb{N}$ is a prime integer, how can i prove that $E_{p^n}$ the group of invertible elements of $\frac{\mathbb{Z}}{p^n\mathbb{Z}}$ is a cyclic group.
0
votes
1answer
58 views

Quote objects in concrete categories

I refer to 'Analogy of ideals with Normal subgroups in groups' which was a very enlightening question for me. When I was young I was too avid on abstract algebra and I did too many courses at the same ...
1
vote
1answer
46 views

Explain rings and is [S, /, -] a ring?

Okay, so we are going to use the base set of numbers [i], which contains all possible cases of ai, where a is any real number. Here are 4 possible groups on this set --> [i,*]... [i,+]... [i,/] ...
1
vote
0answers
46 views

Soluble group algebras and centralizers

Let $K$ be field with $\mathrm{char}(K) = p > 0$ and $G$ a finite group such that $KG$ is soluble. Then the $p$-Sylow-subgroup of $G$ is normal and contains the derived group of $G$ and let $H$ be ...
6
votes
4answers
160 views

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$?

When does $V/\operatorname{ker}(\phi)\simeq\phi(V)$ imply $V\simeq\operatorname{ker}(\phi)\oplus\phi(V)$? I wrote it down in an imprecise way on purpose. The notation above is the linear algebra one: ...
2
votes
3answers
132 views

What is so special about $a*b^{ -1}$ equivalence?

This equivalent is used often in group theory. For example, using this equivalnce you prove Lagranges theroem and also this equivalence gives you the cosets and other things. This equivalence also ...
2
votes
1answer
29 views

Nilpotent and Invertible elements in commutative ring with 1

Let $R$ be a commutative ring with $1$, $S$ a subring also with $1$. Suppose $R\setminus S$ contains a nilpotent element. Prove that $R\setminus S$ also contains an invertible element. Attempt at ...
2
votes
2answers
23 views

Torsion elements in integer-modules

In a worryingly short amount of time I've managed to forget almost everything I knew about modules, groups, rings e.t.c. I'm using the definition that an element $m$ of a module $M$ over an integral ...
1
vote
1answer
55 views

Noetherian group rings

I'm asking for an example of a finitely generated amenable group $G$ and a field $K$, such that the group ring $K[G]$ is not Noetherian. Is it also possible to find a finitely generated amenable ...
0
votes
1answer
33 views

My question is about the definition of a map called the “reduction map”.

Let $G$ be a group and $N$ normal in $G$. I have read about a map $\alpha : G\rightarrow \frac{G}{N}$ called the reduction map mod $N$. I would love if someone could please explain this to me. Is it ...
0
votes
0answers
24 views

In an infinite cyclic field of non zero units, characteristic $\neq 2$, can an element $-u \neq u$ be expressed as $u^t$ for some finite integer $t$?

For the sake of a proof using contradiction ( to be used somewhere), Lets assume that an infinite cyclic field $F$ of non zero units exists with characteristic $\neq 2$ . In this infinite cyclic field ...
23
votes
6answers
2k views

What is Abstract Algebra essentially?

In the most basic sense, what is abstract algebra about? Wolfram Mathworld has the following definition: "Abstract algebra is the set of advanced topics of algebra that deal with abstract algebraic ...
3
votes
3answers
174 views

On the Importance of the Second and Fourth Isomorphism Theorems

I suppose I'd like to focus on the theorems for groups and rings, first of all. In particular, I'd rather not see anything about modules, simply because I don't feel I know enough about them. Anyway, ...
0
votes
1answer
112 views

Subset of $\mathbb{Z} \times \mathbb{Z}$

I have a past exam question that is as follows: Let $k$ be a fixed integer and $S = \{(a,ka)|a \in \mathbb{Z}\}$ be a subset of $\mathbb{Z} \times \mathbb{Z}$. Prove that $S$ is a subgroup of ...
1
vote
0answers
27 views

Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. Is ...
2
votes
2answers
132 views

Applications of groups, rings and modules in life [closed]

I have read calculus. Now I'm reading algebra. What is the application of groups, rings and modules in life? When we study derivations we know several applications. What about groups, rings and ...
1
vote
1answer
34 views

The group morphism of tye ring

Let $(G,+,\cdot)$ $(H,+,\cdot)$ be rings, we suppose that the unites $(G*,\cdot)$ and $(H*,\cdot)$ form groups respectively, for example, the matrix ring $M(n,\mathbb{R})$. There is a group morphism ...
13
votes
3answers
597 views

Why are groups “abelian” but rings “commutative”?

I have never seen, in any text, a ring whose multiplication is commutative being called an "abelian ring", even though this would make perfect sense, because this term would necessarily refer to ...
1
vote
3answers
76 views

Prove that $\{x + y \cdot \sqrt{3} | x, y \in \mathbb{Z} \}$ is a ring (or not)

How to prove that $(R, +, \cdot)$ is a ring (or not), where $R = \{x + y \cdot \sqrt{3} | x, y \in \mathbb{Z} \}$? Update. Is this proof correct? $(R, +)$ is an abelian group: Closure: $a, b \in R ...
0
votes
4answers
87 views

$2 \mathbb{Z}$/$6 \mathbb{Z} $ is isomorphic to $\mathbb{Z_3}$

I am trying to understand how $2 \mathbb{Z}$/$6 \mathbb{Z} $ is isomorphic to $\mathbb{Z_3}$ . So far I understand that: $2 \mathbb{Z}$/$6 \mathbb{Z} $ = { $0+6\mathbb{Z}, 2+6\mathbb{Z}, ...
-2
votes
1answer
63 views

True or False? … [closed]

Let be $G$ a group, if $H$ is normal subgroup of $L$ and $L$ is normal subgroup of $G$, then $H$ is normal subgroup of $G$ Let be $f,g$ in $\mathbb Q[x]$, show that if $gcd(f,g)=d(x)$ then there ...
0
votes
0answers
32 views

Properties of $Hom_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Q} )$ [duplicate]

I have to prove that : 1) $Hom_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Q}) \cong \mathbb{Q}$ as abelian groups 2) $End_{\mathbb{Z}}(\mathbb{Q}) \cong \mathbb{Q}$ as rings What I have done: 1) We can ...
0
votes
0answers
18 views

Properties of $Hom_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Q})$

I have to prove that : 1) $Hom_{\mathbb{Z}}(\mathbb{Q}, \mathbb{Q}) \cong \mathbb{Q}$ as abelian groups 2) $End_{\mathbb{Z}}(\mathbb{Q}) \cong \mathbb{Q}$ as rings What I have done: 1) We can ...
0
votes
1answer
41 views

Proving the existence of unity in $R$, where $R$ is the ring of polynomials over complex numbers with $f(0)=0$.

My line of thought is this: we want to prove that there exists some $h(x)\in R$ such that $g(x)h(x)=g(x)$. Therefore $h(x)=1$ for all $x$. But if $h(x)$ is in $R$, then is it not equal to zero at ...
4
votes
2answers
102 views

How can one visualize a homomorphic mapping.

It has been a year or so studying Group theory and Ring theory. Funnily enough, this is the part where i am able to solve most of the questions of the book quite easily, yet not fully understanding ...
3
votes
1answer
55 views

Classify abelian groups $A$ which are irreducible $End(A)$-modules

Classify abelian groups $A$ which are irreducible $End(A)$-modules. I think i did it for finite abelian group $A$ . A finite abelian group $A$ is irreducible iff order of $A$ a is power of prime. ...
5
votes
2answers
128 views

Generality of rings' abelian group

Let G be an abelian (finite) group. Is there a ring $R$ with $G$ isomorphic to the group $(R,+)$?
0
votes
2answers
31 views

Set or Ring, and group of units?

I have a couple of questions. I understand the axioms needed for a ring. But am confused about a unitary ring? does this just mean its a ring but has to have the unit 1? Also I do not understand ...
7
votes
7answers
206 views

Examples of properties not preserved under homomorphism

An isomorphism indicates that two structures are the same, using different names for the elements. Therefore it's obvious that every (algebraic) property of the first structure must be present in the ...
0
votes
1answer
45 views

How to show $c-b\lt b-a$

The question: Let $G$ be an Arf semigroup and $a\lt b\lt c$ be three consecutive elements in $G$. How to show that $c-b\lt b-a$ and how to show that this is not necessarily the case for every ...
1
vote
1answer
41 views

Is there a way to encode a ring into a group?

Is there a meaningful bijection between tne set of all rings and the set of all groups? Thanks.
0
votes
1answer
179 views

Proving that the unities of a ring form a group under multiplication

I am presented with the following task: Show that if $U$ is the collection of all units in a ring $\langle R, +, \cdot\rangle$ with unity, then $\langle U, \cdot\rangle$ is a group. I am still ...
2
votes
5answers
201 views

how to prove $e^{A \oplus B} = $$e^A$ $\otimes$ $e^B$ where A and B are matrices? (Kronecker operations)

how to prove $e^{A+B} = $$e^A$$e^B$ where A and B are matrices? The operations '+' and '*' are defined such that AI + IB = A+B, where I is the identity matrix. I suppose these are called Matrix ...
0
votes
1answer
31 views

modules finite congenerated are closed under extensions

I have to prove some properties about modules finite cogenerated, I´ve already prove that mmodules finite cogenerated are closed under submodules, finite direct sums, but I can´t see how to prove that ...
2
votes
1answer
147 views

Classify Artinian $\Bbb Z$-modules

How can I classify Artinian $\mathbb{Z}$-modules as Noetherian $\mathbb{Z}$-module? (A $\mathbb{Z}$-module is Noetherian iff it is finitely generated). Any hint will be helpful. I have seen the ...
1
vote
2answers
72 views

Classification of Rings [duplicate]

I am trying to classify rings of order 10. I believe the only possible ring is $\mathbb{Z_{10}}$. Thus I am trying to find a map from my ring $R$ to $\mathbb{Z_{10}}$. The most obvious map $f: R ...
1
vote
2answers
56 views

Identifying a quotient ring.

Consider the Quotient ring $\mathbb{Z}[x]/(x^2+3,5)$. Solution: I first tried to take care of $(5)$ in the above ring. Therefor we can consider $\mathbb{Z_5}[x]/(x^2+3)$. Now and interesting point ...
0
votes
2answers
47 views

Quotient ring understanding

I just have a conceptual question regarding quotient rings and its elements. To get my point across, I will use the following example: Consider the quotient ring $\mathbb Z_5[x]/(x+1)^2$. Since the ...
1
vote
2answers
57 views

Quotient rings and identification

I am trying to identify the following quotient ring using Adding relation method from Artin. The ring is $\mathbb{Z}[x]/(6,2x-1)$. I did find solution to a similar problem on MSE. Hence I tried to ...
2
votes
1answer
73 views

Smallest skew-field containing a non-commutative ring.

Let $R$ be an integral domain and take $D = R - \left\{ 0 \right\}$. The ring $D^{-1}R$ is the smallest field containing $R$ as a subring. Now suppose that I have a non-commutative ring $N$. Suppose ...
3
votes
0answers
47 views

$G \simeq R^{\times}$

What is known about the groups G for wich there exist a unitary ring R, such that $R^{\times} \simeq G$? I can easily prove that The only G cyclic with this property(Edit:and odd order) are those who ...
-4
votes
2answers
227 views

multiplicative group $\mathbb{R}^∗$ of non-zero real numbers is not cyclic. [duplicate]

How will I be able to show that multiplicative group $\mathbb{R}^∗$ of non-zero real numbers is not cyclic?
0
votes
1answer
60 views

How to prove that $J(M_n(R))=M_n(J(R))$?

How to prove that $J(M_n(R))=M_n(J(R))$? Here $M_n(R)$ is the ring of matrices of size $n^2$ over the ring $R$. And $J(M_n(R))$ is a two-sided ideal of the ring $M_n(R)$.
5
votes
2answers
592 views

Interview preparation for Ph.D admission

I have recently gave a written exam for admission to Ph.D program to an institute in India. I have done that exam well and hoping for an interview call. I would like to know what could be type of ...
5
votes
2answers
93 views

if $A^\times $ is a commutative group, does $A$ necessarily be a commutative ring?

Let $A$ be a ring and $A^\times$ be the collection of unit elements of $A$. If $A$ is a commutative ring, then $A^\times$ is a commutative group. Conversely, if $A^\times $ is a commutative group, ...
4
votes
1answer
57 views

Submonoid: Identity

There's a common mistake appearing in quite many books in definitions and proofs as well; now my idea is maybe you can safe the day: Imagine you're having a monoid and a subset being closed under the ...
0
votes
1answer
23 views

Identifying some cyclic subgroup

Is there a fast way to argue that (for $a,b>1$ integers) the set of all $x\in\mathbf{Z}/b\mathbf{Z}$ with $ax=0$ is isomorphic to $\mathbf{Z}/{gcd(a,b)}\mathbf{Z}$? Maybe by counting the elements, ...