Abstract algebra is the study of algebraic objects. Some of the more common algebraic objects are groups, rings, fields, vector spaces, modules, among other topics.

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Are ideals also rings?

I am learning about rings and ideals. But I am confused about something. My book (Gallian) says that an ideal of a ring by definition is a subring. But I have talked to other people who insist that an ...
2
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1answer
36 views

Show that if $R$ is commutative, then $\mathrm{Ann}(n_1) + \mathrm{Ann}(n_2) = R$.

Let $R$ be a commutative ring with $1$, and let $M$ be a cyclic $R$-module with generator $m$ (so that $M=Rm$). Suppose that $M = N_1 \bigoplus N_2$ for some submodules $N_1$ and $N_2$ of $M$. Let ...
4
votes
3answers
60 views

Example of a ring where all but two of its elements are units

One way of viewing a field is just as a ring where all but one of its elements (namely $0$) is a unit. I'm looking for rings (commutative with a 1) where all but two of its elements are units. I found ...
1
vote
1answer
29 views

Why is the $\mathbb Z$-module $\mathbb Q$ projective in category $\sigma[\mathbb Q]$?

We know that $\mathbb Q$ is not a projective $\mathbb Z$-module, but, I've read this paper and it says in Example 2.11 that it's easy to check that "$\mathbb Q$ as $\mathbb Z$-module is projective ...
0
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0answers
18 views

Prove if a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is two-sided and is also the unique maximal right ideal in $R$.

If a non-trivial ring $R$ has a unique maximal left ideal $J$ , then $J$ is two-sided and is also the unique maximal right ideal in $R$. I can prove that it is two sided, but I can't prove that it is ...
0
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1answer
25 views

An element in $\mathbb{Q}(c)$ where $c$ is a root.

Let $c$ be a root of $f(x) = x^3 + 4x^2 - 6x + 2.$ An element of $\mathbb{Q}(c)$ can be expressed uniquely in the form $a_2c^2 + a_1c + a_0$ for some $a_2,a_1,a_0 \in \mathbb{Q}$. If we express $c^4$ ...
4
votes
1answer
21 views

Why is the trace submodule of $FS_n$ the unique $1$-dimensional submodule?

Suppose $V$ is an $n$-dimensional vector space over a field $F$, with basis $e_1,\dots,e_n$. Then $S_n$ acts on $V$ by the action $\sigma\cdot e_i=e_{\sigma(i)}$, and $V$ is a $FS_n$-module, where ...
0
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2answers
23 views

Relation between reduced finite algebra, prime ideal and field extension

Is it true that if $L$ is a reduced finite dimensional commutative algebra over a field $K$ (which is finite or of characteristic $0$) and if $\mathfrak p$ is a prime ideal of $L$, then ...
2
votes
0answers
42 views

Prove $\phi$ is a homomorphism.

Let H and K be normal subgroups of a group $G$ with $H \subseteq K$. Define $\phi: G/H \rightarrow G/K$ by $\phi(Ha)=Ka$ Prove $\phi$ is a homomorphism. We are given a function $\phi$, to prove ...
2
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1answer
28 views

Prove that if $M$ is a simple left $R$-module, $Ann(m)$ is a maximal left ideal of $R$ and $M \cong R/Ann(m)$ for all $m \in M\backslash \{0\}$

I need to prove equivalent statements of a simple left $R$-module. However, I'm stuck at the following: The map $\phi_m: R/Ann(m) \rightarrow M$, such that $\phi_m(r+Ann(m)) = rm$ for all $r + ...
0
votes
0answers
33 views

Find all non-trivial submodules of a direct sum of two non-isomorphic simple modules

Let $R$ be a ring with $1$. Let $M_1$ and $M_2$ be two non-isomorphic simple (nonzero) $R$-modules. Find all non-trivial submodules of $M_1 \bigoplus M_2$. Solution: $M_1 \bigoplus M_2 \cong M_1 ...
1
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0answers
33 views

A quick way to determine number of ring homomorphism

Find the number of ring homomorphism from $\mathbb Z_2\times \mathbb Z_2$ to $\mathbb Z_4$. I know that checking idempotent elements the number of ring homomorphisms is $1$. Again the number of ring ...
3
votes
2answers
41 views

If $x = a +bi$ is not a unit of $\mathbb{Z}[i],$ prove that $a^2+b^2>1.$

A problem from my algebra text: If $x = a +bi$ is not a unit of $\mathbb{Z}[i],$ prove that $a^2+b^2>1.$ I think it's false since $x = 0 + 0i = 0 \in \mathbb{Z}[i]$ is not a unit, but $0 + 0 ...
1
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1answer
34 views

Let $R$ be a non-commutative ring. Show that if $R$ is simple and has 1, then $Z(R) = \{a \in R | ra = ar$ for all $r \in R \}$ is a field.

Let $R$ be a non-commutative ring. Show that if $R$ is simple and has 1, then $Z(R) = \{a \in R | ra = ar$ for all $r \in R \}$ is a field. I think what I need to do is to show that $Z(R)$ is simple ...
2
votes
2answers
30 views

In $\Bbb Z[\sqrt{2}]=\{a+b\sqrt{2}\rvert a,b∈\Bbb Z\}$, show that every element of the form $(3+2\sqrt{2})^n$ is a unit, where n is a positive integer

In $\Bbb Z[\sqrt{2}]=\{a+b\sqrt{2}\rvert a,b∈\Bbb Z\}$, show that every element of the form $(3+2\sqrt{2})^n$ is a unit, where n is a positive integer. My understanding of a unit is that if a ...
0
votes
1answer
37 views

Quotient of Ideals in matrix rings

I'd like to know where could I find some info about the quotient $I:J=\{a\in R\mid aJ\subseteq I\}$ ($R$ a ring) in matrix rings? Or for example, in a matrix ring over $\mathbb{Z}$. I would like to ...
0
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0answers
22 views

Let $E$ be an extension of $F$ and let $a,b\in E$. Prove that $F(a,b)=F(a)(b)=F(b)(a)$.

Let $E$ be an extension of $F$ and let $a,b\in E$. Prove that $F(a,b)=F(a)(b)=F(b)(a)$. My thought process is to assume $a,b\in F$, then use the associative property of multiplication. Seems ...
-2
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1answer
38 views

Could someone explain in basic terms what an “algebra” means? [on hold]

I've seen the terms "lie algebra" and "algebra over a field" and I'd like to know what they mean. I have very basic knowledge of group theory if that helps.
1
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1answer
38 views

Proof of Newton Girard formula symmetric polynomials

Newton Girard formula states that for $k>2$: \begin{equation} p_k=p_{k-1}e_1-p_{k-2}e_2+\cdots +(-1)^{k}p_1e_{k-1}+(-1)^{k+1}ke_{k} \end{equation} where $e_i$ are elementary symmetric functions and ...
2
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1answer
26 views

How to classify the rings of fractions of a principal ideal domain?

Let $A$ be a principal ideal domain and let $K$ be its field of fractions. I proved a) Every ring $B$ such that $A \subset B \subset K$ is a ring of fractions of $A$, and c) Show that any ring of ...
0
votes
2answers
31 views

$x^{p^{k}-1}-1$ divides $x^{p^{n}-1}-1$ in $\mathbb{F}_{p}$ iff $k$ divides $n$

Let's consider two polynomials in $\mathbb{F}_{p}[x]$: $f(x)=x^{p^{n}-1}-1$ and $g(x)=x^{p^{k}-1}-1$. How to prove that $g(x) \mid f(x)$ iff $k \mid n$? For instance, it's worth trying this: ...
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4answers
71 views

How to prove this is a field?

Let $F=(\mathbb{Z}/5\mathbb{Z})[x]/(x^2+2x+3)$. How do I prove $F$ is a field? I've shown its a commutative ring with an identity $\bar1$. Then we let ...
2
votes
0answers
29 views

Determining all the subfields of the splitting field for $x^8-2$ which are Galois over $\mathbb{Q}$

I have found all the subgroups of the Galois group of the splitting field $K=\mathbb{Q}(\sqrt[8]{2},i)$ over $\mathbb{Q}$, and I know that if any of the subgroups $H$ of $G=$Gal$(K/\mathbb{Q})$ are ...
0
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2answers
34 views

Let $R$ be a PID and $I$ a prime ideal of $R$ s.t. $0 \subset I \subset 1_R$

Let $R$ be a PID and $I$ a prime ideal of $R$ s.t. $0 \subset I \subset 1_R$ and let $I = \langle a \rangle$, where $a$ is a prime element of $R$. My question is: is there any other prime ideal $J$ ...
0
votes
1answer
18 views

Let G be a group. Use the FHT (Fundamental Homomorphism Theorem) to prove that the quotient group G/{e} is isomorphic to G.

Ok so the Fundamental Homomorphism Theorem (or First Group Isomorphism Theorem) states that if θ : G → H and ker (θ) = K, then the quotient group G/K is isomorphic to H. I know that θ : G → G has to ...
1
vote
1answer
34 views

Prove that $\Bbb{Z}[i]/I$ is finite where I is an ideal of $\Bbb{Z}[i]$

Show that for any nontrivial ideal $I$ of $\Bbb{Z}[i]$, $\Bbb{Z}[i]/I$ is finite. $\Bbb{Z}[i]$ is a PID, so $I=\langle{a+ib\rangle}$. Now $\Bbb{Z}[i]/I$ has elements of the form ...
1
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2answers
206 views

Can a non-commutative ring contains identity?

Can a non-commutative ring $R$ contains identity? Suppose $R$ contains the identity element 1. Construct an ideal $Z(R) = \{a \in R \mid ra = ar\text{ for all }r \in R\}$. Since $1 \in Z(R)$, $R = ...
0
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0answers
24 views

Proof of primitive element theorem for $F$ finite

I am trying to understand the proof of primitive element theorem, in particular this statement: Let $E/F$ be a field extension such that there are finitely intermediate fields containing $F$. Then $E ...
0
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0answers
25 views

Can icosahedral group have subgroup of order 30?

The textbook (Artin) wants me to show that there is no subgroup of order 30 in the icosahedral group. I tried to use the index(2) of the subgroup of order 30, but I can't get any new ideas. What other ...
0
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1answer
32 views

Modules over Itself

Let $F$ be a field and let $R$ be the ring of polynomials over $F$ in infinitely many variables $x_1, x_2 , \cdots x_n \cdots$ (Of course, each element of $F$, being a polynomial, will involve only ...
1
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1answer
30 views

How do I know an element generates a coordinate ring K[W] as a vector space over K?

I have an example which proves that a cuspidal cubic $W\subset \mathbb{A}^2$ defined by $y^2-x^3=0$ is not isomorphic to to $\mathbb{A}^1$. I'll start by defining a few things: Let $V=\mathbb{A}^1$, ...
2
votes
2answers
50 views

Direct sum of two non-zero $R$-modules

If $R$ is a commutative integral domain, how can I show that it cannot decompose as a direct sum of two non-zero $R$-modules (when viewed as a module over itself). If $n\geq 1$, is there an example ...
0
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0answers
51 views

Prove $\mathbb{Z} [\sqrt{2}]$ is a Euclidean ring.

The definition of Euclidean ring: An integral domain R is called Euclidean ring if $\exists \delta$ : $R${$0$} -> $\mathbb{N} \cup{0}$ satisfying: (1) $\delta (a) \leqslant \delta (ab)$ if a, b $\in ...
1
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1answer
35 views

For a $p$-group $G$, and $H \le G$, if $G=H G^\prime$, then $H=G$ [on hold]

For a $p$-group $G$, and $H \le G$, prove that, if $G=H G^\prime$, then $H=G$; where $G^\prime$ is the commutator subgroup of $G$.
0
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1answer
112 views

Is this element algebraic over $\mathbb{Q}$?

I'm trying to see whether the following element is algebraic over $\mathbb{Q}$ and if it is - try and find its degree: $$ a = \left(\frac{(1 + \sqrt[3]{7})^{7/5}}{(\sqrt[3]{7} - 7)^3 + ...
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0answers
19 views

Subgroup $H \leq G$ acting on $G$ by translation is transitive?

In Elementary Topology. Textbook in Problems, by Viro, et al they state the following: Let $G$ be a topological group, $H \leq G$ a subgroup. Then $G$ is a homogeneous $H$-space under the ...
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2answers
31 views

Example such that $ f_1(x)$ is reducible but $f(x)$ is irreducible.

(The (mod p) Irreducibility Test) Let $p$ be a prime an suppose that $f(x) \in \mathbb Z[x]$ with $\deg f(x) \geq 1$. Let $f_1(x)$ be the polynomial in $\mathbb Z_p[x]$ obtained from $f(x)$ by ...
2
votes
1answer
48 views

Show that the free product of countably many countable groups is countable.

The question I am struggling with is as follows: Suppose that $\{G_\alpha\}$ is a countable collection of countable groups. Show that $\ast_{\alpha}G_\alpha$ is countable. The definition of ...
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0answers
24 views

Conjugacy classes of solvable groups [on hold]

If $A$ be a subset of solvable group $G$, let $k_G(A)$ be the number of $G$- conjugacy classes contained in $A$. Also let $N$ be a normal subgroup of $G$ of odd order, $\frac{G}{N}$ be an abelian ...
0
votes
1answer
25 views

Prove that if G is finite, then |H ∩ K| is a divisor of |H| and a divisor of |K|. [on hold]

H and K are subgroups of a group G, and H ∩ K is a subgroup of H and a subgroup of K ===Prove that if G is finite, then |H ∩ K| is a divisor of |H| and a divisor of |K|. =====If |H| = 28 and |K| = ...
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0answers
16 views

Help in the proof of $k \subset F$ is radical if and only if it is solvable.

Let $k \subset F$ be a Galois Extension, with char $k=0$. Provided it has enough roots of $1$, $k \subset F$ is radical if and only if it is solvable. I am trying to show that Solvability $\implies$ ...
2
votes
1answer
37 views

Primitive element and Galois group Computation

Let $L=\mathbb{Q}(\sqrt{2},\sqrt{3},y)$ where $y^2=(9-5\sqrt{3})(2-\sqrt{2})$. Show that $y$ is a primitive element of the extension $L/\mathbb{Q}$. Determine its minimal polynomial and a splitting ...
1
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1answer
58 views

Writing equations without latex (for labelling, classification, structure analysis and rendering according to mathematical meaning)

I'm searching a solution to write equations other than (but compatible with) latex. While latex is excellent for math rendering, it is not suited to describe the structure or meaning of the equations. ...
5
votes
1answer
50 views

Finite conjugate subgroup

In a paper titled "Trivial units in Group Rings" by Farkas, what does it mean by Finite conjugate subgroup. Here is the related image attached- What is finite conjugate subgroup of a group? It is ...
0
votes
1answer
24 views

Construct the Normal Closure for this extension

I am trying to find the normal closure for the following extension $\Bbb Q\subset\Bbb Q(t)$, where "$t$" is a zero of $x^3-3x^2+3$ and $\Bbb Q$ are the rational numbers. I know the normal closure ...
2
votes
2answers
44 views

If K and H are normal subgroups of $G$, $H \cap K = \{1\}$ and both $G/H$ and $G/K$ are abelian, then $G$ is abelian.

Let G be a group, and $H \trianglelefteq G$, $K \trianglelefteq G$. Prove that if $H \cap K = \{1\}$ and $ G / H $ and $ G/ K $ are abelian, then G is abelian. I've tried to give a proof by ...
0
votes
1answer
15 views

$A$ prime in $S$ implies that $\phi^{-1}(A)$ prime in $R$ ; $A$ maximal in $S$ implies that $\phi^{-1}(A)$ maximal in $R$

Suppose $R,S$ are commutative rings with unities. Let $\phi$ be a ring homomorphism mapping $R\to S$ and let $A\subset S$ be an ideal. How can I start the proofs for: Showing that $A$ prime in ...
0
votes
1answer
14 views

Logical comparison of two values with algebra

Suppose I have two real numbers A and B (A $\wedge$ B $\subset$ $\mathbb{R}$). I want to do some algebra over these number and get 1 if they are equal and get 0 if not. For example: In this ...
0
votes
0answers
20 views

Ordering rational functions in a field

This is from Fraleigh, A First Course in Abstract Algebra, 7th edition, Section 25, "Ordered Rings and Fields", questions 10, 11,12, and 13. Here's the question: List the given elements in an order ...
1
vote
2answers
25 views

If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$.

Herstein 3.4.20: If $R$ is a ring with unit element $1$ and $\varphi$ is a homomorphism of $R$ onto $R'$, prove that $\varphi(1)$ is the unit element of $R'$. I don't understand why $\varphi$ needs ...