This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.

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1answer
26 views

If $\cap_{j=1}^{n}I_{j} \subseteq P$ for any ideals $I_1,I_2,..I_n$, then $I_j \subseteq P$ for some $j$

$P$ is a prime ideal if $P$ satisfies the following : If $\bigcap\limits_{j=1}^{n}I_{j} \subseteq P$ for any ideals $I_1,I_2,..I_n$, then $I_j \subseteq P$ for some $j$, where $R$ is a commutative ...
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2answers
61 views

Sum and product of comaximal ideals

Let $R$ be a commutative ring with unity. If $R=I_{i}+I_{j}$, for all $i\ne j$, where $I_1,I_2,...,I_n$ are ideals of $R$, I want to show that $$R=I_{n}+I_{1}I_{2}\cdots I_{n-1}.$$ I started off ...
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1answer
28 views

Questions related to the concept of $k$-algebras

I am reading about modules and some days ago I've worked on some exercises related to $k$-algebras. The definition I've seen of $k$-algebra is that it is a field $k$ and a ring $A$ together with a ...
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0answers
12 views

Is a field a PID? [duplicate]

How can I prove that a field is a PID? I can prove that a field is an Integral Domain, but stuck in proving that every ideal is principal.
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0answers
56 views

Quotient Field of an Integral Domain

The question is: Let $n$ be a positive integer and let $D$ be the subset of all rational numbers of the form $$\frac{a}{n^k}$$ with $a \in \mathbb{Z}$ and $k$ any positive integer. Show that $D$ ...
1
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1answer
53 views

Some questions about finite rings [closed]

Let $K$ be finite associative ring with nonzero multiplication. Are the following statements true: If for an element $a \in K$ there exist an element $b \in K$ such that $ax=b$ for all nonzero ...
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0answers
39 views

Surjective ring homomorphism from $M_n(R)$ to $M_n(R/I)$ where $R$ is a ring and $I$ is an ideal for R?

I'm looking for such a surjective homomorphism. I was thinking of starting from the canonical surjection from $R$ to $R/I$ and induce one but somehow I get stuck... Can you help me? Thank you very ...
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2answers
48 views

The krull dimension of $\Bbb{Z}$ and artinian rings

On page thirty of Matsumura, it says that $\Bbb{Z}$ has krull dimension 1 because every prime ideal is maximal. I understand this because for any prime p you have $0 \subset p$. However, for artinian ...
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3answers
54 views

Show some polynomial is irreducible over the field of 7 elements.

I have to show that the polynomial $x^4+x^3+x^2+x+1$ is irreducible over the field $F_7$. It doesn't have roots in $F_7$, but I can't show it does not have degree two irreducible factors in $F_7[x]$. ...
3
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1answer
40 views

Let R be a finite commutative ring with 1.Let a,b∈ R such that (a)+(b)=R and M be any maximal ideal of R with b∉M. Then is some x∈ R such that a+xb∉M

Let $R$ be a finite commutative ring with $1$.Let $a , b\in$ $R$ such that ($a$)+($b$)=$R$ and $M$ be any maximal ideal of $R$ with $b\not\in M$.Then I have to prove that there is some $x\in$ $R$ such ...
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1answer
19 views

What exactly does it mean for a maximal ideal to be unique in a principal ideal domain?

I'm currently reading about PIDs and have come across a question involving maximal ideals which at one point reads "Suppose that a Euclidean domain $R$ had a unique maxima ideal $P$". Does this mean ...
1
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1answer
38 views

The ring Z/pZ[Z/pZ] (p is a prime) is not a semi-simple ring.

A ring A is semi-simple if and only if it is Artinian and radical(intersection of all of its maximal ideal) of A is zero. As the ring Z/pZ[Z/pZ] is finite,so it is Artinian.So now it is enough to show ...
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1answer
15 views

Morphism of $k$-algebras between abelian group of $n \times n$ matrices and $m \times m$ matrices

Problem Let $k$ be a field and $f:M_n(k) \to M_m(k)$ be a morphism of $k$-algebras ($n,m \in \mathbb N$). Prove that $n$ divides $m$. I have no idea what to do here, I thought that maybe I should ...
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0answers
21 views

Show that there exists a prime ideal $P$ of $R$ such that $ I \subseteq P$ and $P \cap S=\emptyset$ [duplicate]

Show that there exists a prime ideal $P$ of $R$ such that $ I \subseteq P$ and $P \cap S=\emptyset$, if $S$ is multiplicatively closed subset of $R$, $I$ is an ideal of $R$ such that $I \cap S = ...
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0answers
48 views

Prove that the intersection of all maximal left ideals of a ring $R$ is a two sided ideal

Prove that the intersection of all maximal left ideals of a ring $R$ is a two sided ideal. What i did:Suppose $B$ be the intersection of all maximal left ideals of the ring $R$. Clearly $B$ is a left ...
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1answer
37 views

Prove that $M_{n\times n}(K)$ and $P_{n^2-1}[x]$ are not isomorphic rings

Prove that $M_{n\times n}(K)$ and $P_{n^2-1}[x]$ (polynomials with degree less than or equal to $n^2-1$) are not isomorphic rings for any field $K$ and $n\ge 2$ Let $f: M_{n\times n}(K)\to ...
2
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1answer
69 views

Give an example of a maximal ideal in a noncommutative ring which is not prime

While trying to find an example, I came up with this: Since if $J$ is an ideal of a the ring $M_n(R)$, where $R$ is a commutative ring, then $J=M_n(I)$ for some ideal $I$ of $R$. IF I could show that ...
0
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1answer
22 views

Prove that a polynomial an irreducible g has no multiple root in C

I was looking at a question from Artin from Algebra which says that an irreducible polynomial g in F[x] where F is subfield of $\mathbb{C}$. So as per the proofs I have seen so far says as - ...
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1answer
61 views

Why does Proposition 1.8 in Atiyah-Macdonald imply that the smallest prime $\mathfrak{p}$ containing a primary ideal is equal to its radical?

Proposition 4.1 in Atiyah-Macdonal states that the radical of a primary ideal is the smallest prime ideal containing the primary ideal. They start the proof claiming that showing the radical is a ...
0
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1answer
47 views

Suppose that $K$ is a field and that $f$ and $g$ are relatively prime in $K[x]$. Show that $f - Yg$ is irreducible in $K(y)[x]$.

I'm a bit confused of the notation $K(y)[x]$, is that simply $K[y][x]$ so... $K[y,x]?$ Anyways, here's my attempt at trying this before I get stuck. Since $f$ and $g$ are relatively prime, that ...
2
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1answer
56 views

Zerodivisors in polynomial rings over a non-commutative ring

Prove that if $f \in R[x]$ is a zero divisor then $\exists r(\neq 0) \in R$ s.t $rf=0$, where $R$ is a ring. I know that for $(a_0+a_1x+ \cdots ...
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4answers
57 views

Justify: $R$ is commutative

Let $R$ be a ring. If $(ab)^n=a^nb^n$ holds for all $n$ where $a, b\in R$ then justify: R is commutative ring. No idea how to prove it. But also fail to get one counter example. Please help thanks ...
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0answers
100 views

Hilbert's Basis Theorem - Clever Proof?

So I am studying commutative algebra at the moment and I have come across the proof of the Hilbert Basis Theorem (the proof I have is the same as the one in Reid's "Undergraduate Commutative ...
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0answers
16 views

Unique maximal ideal in the ring of fraction

Let $R$ be a commutative ring with 1, and $P$ be a prime ideal in $R$. Let $D = R$ \ $P$. Show that $R_P := D^{-1}R$ has only one maximal ideal. Problem 2b in this link ...
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1answer
487 views

Why are there so many universal properties in math?

I don't really understand why there are so many universal properties in math or why they all need to be highlighted. For example, I'm studying some Algebra right now. I have found three universal ...
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2answers
138 views

Vector space as simple $K[x]$-module

I am trying to solve the problem: Let $V$ be a vector space and $T$ a linear transformation $T:V \to V$. Let $(V,T)$ be a $K[x]$-module. Show that $(V,T)$ is simple if and only if $V$ is finite ...
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1answer
36 views

Prove there exists a unique polynomial

I'm having trouble proving the following lemma: Let $ p $ be a prime and $ f \in \mathbb{Z}_p [x_1, \dots, x_n] $. Prove there exists a unique polynomial $ f^* \in \mathbb{Z}_p [x_1, \dots, x_n] $ ...
2
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1answer
29 views

Constructing noncommutative nilpotent rings of given index

When I read about algebra I often see a certain disregard for examples or perhaps a disregard for a reader whose knowledge of examples is limited. When I'm interested in a property $p$ of an algebraic ...
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1answer
49 views

If $k[X]/f = k[X]/g$, does $f = g$?

Let $k$ be a field and $f, g$ be irreducible monic polynomials in $k[X]$. Suppose $k[X]/f \stackrel{\sim}{=} k[X]/g$. Then does $f = g$? If so, how can this be generalized? Otherwise, how should I ...
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2answers
64 views

$M$ is a simple module if and only if $M \cong R/I$ for some $I$ maximal ideal in $R$.

I am trying to show the following statement (taken from Rotman's Advanced Modern Algebra): Let $M$ be an $R$-module. Then $M$ is a simple module if and only if $M \cong R/I$ for some $I$ maximal ...
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2answers
42 views

Every element in a ring different from 0 is invertible [closed]

True or false.All number r in a ring R, different from 0 is same. It seems like it is true but how to go about proof if we consider Matrix then it is invertible if determinant is not 0 .
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3answers
67 views

Commutative subring of non-commutative ring [closed]

How to find a commutative subring of a non-commutative ring where $R=M_n(\mathbb{Z}_r) $ where $r$ is a prime and $n\geqslant2$? Thanks
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0answers
37 views

Maximality of the kernel of a quasifield

Setting A (left) quasifield is an algebraic structure $(Q,+,\cdot)$ such that $(Q,+)$ is an abelian group. (As usual, we denote its identity element by $0$.) Each equation $x\cdot a = b$ with ...
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1answer
26 views

Find the remainder of $(p-2)!$ module $p$, where $p$ is a prime $\geq 3$

My attempt: From Wilson's Theorem: For a prime $p$, $$(p-1)! \equiv (-1) \pmod p$$ Multiplying both sides by $(p-2)$, $$(p-2)! \equiv -(p-2) \pmod p$$ i.e. $$(p-2)! \equiv 2 \pmod p$$ So the ...
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2answers
37 views

How to see a polynomial in the quotient ring $\mathbb{Z} / 3\mathbb{Z} $ [closed]

In the quotient ring $\mathbb{Z}/ 3\mathbb{Z} $ what does a polynomial look like? Specifically, what is the common factor of $x^2 +x+1$ and $x^3+x+1$?
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0answers
24 views

the sum of two $z^0$-ideals even in $C(X)$ need not to be a $z^0$-ideal

I need to get an example of two $z^0$-ideals while their summation is not? What i know that the sum is a $z^0$-ideal or all of $C(X)$ if and only if $X$ is quasi F-space So i'm searching for an ...
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1answer
129 views

The ring of idempotents

Let $R$ be a commutative ring. Then its ring of idempotents $I(R)$ consists of the idempotent elements of $R$, with the same multiplication as in $R$, but with the new addition $x \oplus y := ...
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2answers
41 views

Find the remainder of $49!$ modulo $53$

Since $53$ is prime, from Wilson's theorem, $52! \equiv -1\pmod{53}$, i.e. $52 \times 51 \times 50 \times 49! \equiv -1\pmod {53}$ I don't understand how to take it from here. The other form I ...
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1answer
21 views

Solve $22x \equiv 5(mod 15)$

I looked at an example of this type, and here's my attempt: $gcd(22,15)=1$ and $1$ is a divisor of $5$ so solutions exist. Now $22x \equiv 5(mod 15)$ is the same as solving $22x=5$ in $Z_{15}$ i.e. ...
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1answer
39 views

Compute the remainder of $2^{(2^{17})}+1$ when divided by $19$

Compute the remainder of $2^{2^{17}}+1$ when divided by $19$ Hint given in book: Computer the remainder of $2^{17}$ modulo $18$ My attempt: From Fermat's little theorem, $2^{18}=1(mod19)$ I have ...
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1answer
33 views

Problem related to modules and k-algebras

I am trying to do the following exercise: Prove that if $A$ is a $k-$algebra and $M$ is a module then the product $\lambda * m=\tau(\lambda)m$ ($\tau$ is the morphism from $K$ to $Z(A)$) defines on ...
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2answers
82 views

Finding all primitive polynomials of a certain degree in $\mathbb{F}_q$

I am writing an algorithm to find all primitive polynomials in $\mathbb{F}_2[X]$ and I found this theorem : If $P(X)$ is a primitive polynomial in $\mathbb{F}_p[X]$ of degree $n$ with root $a$, then ...
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3answers
35 views

Find the characteristic of the ring $\mathbb Z_6 \times \mathbb Z_{15}$

My attempt: Let the characteristic be $n$. Then, $n \cdot (1_6, 1_{15}) = (0_6, 0_{15})$, i.e. $n \cdot 1_6=0_6$ and $n \cdot 1_{15}=0_{15}$ The least $n$ for which both are true is $30$, so $30$ ...
0
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1answer
23 views

Solve the equation $3x=2$ in the fields $\mathbb Z_7$ and $\mathbb Z_{23}$

This is a sum from Abstract Algebra by Fraleigh. Myy attempt: $$3x=2$$ $$\Rightarrow 3x-2=0$$ Now, the elements of $\mathbb Z_7$ are {$0,1,2,3,4,5,6$} Substituting these values in the left side ...
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1answer
30 views

A simple module

Suppose $R$ is a ring and $M$ is an $R$-module. Prove: $M$ is simple if and only if there is a left maximal ideal $m$ such that $M\cong R/m$.
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0answers
16 views

Polynomial rings, units, nilpotent elements [duplicate]

Assume R is commutative Let $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ be an element of the polynomial ring $R[x]$. Prove that $p(x)$ is a unit in $R[x]$ iff $a_0$ is a unit and $a_1,a_2,...,a_n$ are ...
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0answers
17 views

Prime Ideals and Maximal Ideals in Euclidean Domains

Prove that every nonzero prime ideal in a euclidean domain is maximal. This is what I have so far: Let R be a euclidean domain and let P be a nonzero prime ideal in R generated by a. So, P=(a) and ...
0
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0answers
32 views

Semidirect product of rings?

I'm trying to understand a paper. We are given a $\mathbb{C}[m^\pm, l^\pm]$-module $M$. Then we want to extend the module structure to $\mathbb{C}[m^\pm, l^\pm]\rtimes \mathbb{Z}_2$. The action of $s ...
2
votes
2answers
38 views

$|R|=30$ and $|I|=10$ then $I$ is a maximal ideal

How shall I check this: Suppose that $R$ is a commutative ring and $|R|=30.$ If $I$ is an ideal of $R$ and $|I|=10$ show that $I$ is a maximal ideal. Please give some hint how to go with solution...
1
vote
1answer
48 views

If $R$ is a ring and $A$ is a maximal ideal of $R$ then $R/A$ is a field

I've to prove one way : If $R$ is a ring and $A$ is a maximal ideal of $R$ then $R/A$ is a field. Now suppose that $A$ is maximal and let $b \in R$ but $b \notin A$. It suffices to show that $b+A$ ...