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
20 views

I is equal to the preimage of its image.

Lemma. Let $f$ be a homomorphism from the ring $R$ onto the ring $R'$. If $I$ is any ideal of $R$ such that $\ker(f)$ is a subset of $I$, then $I = f^{-1}(f(I))$. I am trying to understand this ...
3
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1answer
60 views

Isomorphic or equal?

Let $\sim_n$ be the usual equivalence relation of congruence modulo $n$ in $\mathbb{Z}$, i.e., for $a,b\in\mathbb{Z}$, $a\sim_nb\Leftrightarrow a-b=k\cdot n$ for some $k\in\mathbb{Z}$. For $n=0$ the ...
0
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4answers
49 views

If $I$ and $J$ are distinct ideals in ring $R$ and $f:R \to R'$ is a homomorphism then is $f(I) = f(J)$?

The text book I am reading says that if $I$ is a subset of $J$ and $J$ is a subset of $I + \ker (f)$ then $f(I) = f(J)$. The argument goes: $f(I)$ is a subset of $f(J)$ is a subset of $f(I + \ker (f)) ...
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0answers
29 views

Proof that the kernel is a normal subgroup of the domain: repeated line

On proofwiki (https://proofwiki.org/wiki/Kernel_is_Normal_Subgroup_of_Domain), the lines corresponding to 'definition of identity' and 'definition of kernel' are identical. Why do we need the second ...
2
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1answer
27 views

is $\mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \{1 \}$ divisible subgroup of $ \mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \mathbb{Z}_2$?

According to Unit Groups of Classical Rings by Karpilovsky, p.107 we know that: If $F$ is a real-closed field, then $F^*‎\simeq‎ \mathbb{Q} ^{|F|} ‎\times‎ \mathbb{Z}_2$. Now, we know that ...
3
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1answer
44 views

A ring isomorphic to its square [duplicate]

Is there an example of a ring $A$ (with unity) which is isomorphic as unital rings to $A\times A$? Any such ring can't have invariant basis number so in particular can't be commutative.
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0answers
31 views

Graphs associated with rings and modules

There are several articles in the literature that deals with some interesting graphs associated with rings and modules. For example The zero-divisor graphs D. F. Anderson, P. S. Livingston, The ...
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1answer
58 views

Prove that $f$ is a nonzerodivisor on $R[x_1,\dots,x_r]/IR[x_1,\dots,x_r]$ for every ideal $I$ in $R$

Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ be a nonzerodivisor of $S$. Suppose that the ideal generated by the coefficients of $f$ is $R$. How to ...
2
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2answers
43 views

Show that the ideal $I=\left\langle x_1^2+1,x_2,…,x_n\right\rangle$ is maximal in $\mathbb{R}[x_1,…,x_n]$.

This is an exercise in "Ideals, varieties, and algorithms" by Cox et al. It first asks to show that $I=\left\langle x^2+1\right\rangle$ is maximal in $\mathbb{R}[x]$. I can show it because it is a ...
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2answers
31 views

Find character table for symmetric group $S_3$

This group contains all permutations of 3 elements, so it has order 3!=6. Its three congruency classes are {1}, {(1,2),(1,3),(2,3)}, {(123),(132)}. As we know that the number of congruency classes ...
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4answers
50 views

$\mathbb Z_n$ is semisimple iff $n$ is square free

$\mathbb Z_n$ is J-semisimple iff $n$ is square free. A ring $R$ is said to be $J$ semisimple if intersection of all maximal ideals of $R$ is $\{0\}$. If $n$ is square free then ...
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1answer
70 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
3
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0answers
24 views

Why is this done? (Quadratic integer rings definition) [duplicate]

From Dummit & Foote pg. 229: Let $D$ be a squarefree integer. It is immediate from the addition and multiplication that the subset $\Bbb{Z}[\sqrt{D}] = \{a + b \sqrt{D} | a,b, \in \Bbb{Z}\}$ ...
1
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1answer
32 views

Quasi Injective vs Pseudo Injective modules

$\textbf{Definition:}$ A left $R$ module $M$ is called QI (PI) - module if for every submodule $N$, any $R$-homomomorphism (monomorphism) $N\rightarrow M$ extends to an endomorphism of $M$. For QI - ...
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0answers
50 views

Uniqueness of the decomposition of an ideal

Let $ F $ be a non-empty subset of $ \{ 1,2,\dots,n\} $ and $ P_{F}=(\{x_{i}:i\in F\}) $. Let $ F_{1},F_{2},\dots,F_{m} $ be pair-wise distinct non-empty subsets of $ \{1,2,...,n\} $ and $$ ...
2
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1answer
102 views

$R$ is normal. Are $R[x]$ and $R[[x]]$ normal?

Studying about normalizations I've bumped in the following theorem: Theorem. Let $R$ be a normal (integrally closed) domain, then $R[x]$ is a normal domain. How to prove (elegantly, if possible) ...
0
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1answer
11 views

What conditions make the ring of Laurent polynomials in non-commuting variables countable?

Suppose we have some commutative ring $R$ and the ring of Laurent polynomials in a finite number of non-commuting variables $S=R\langle x_1,\,x_1^{-1},\ldots,\,x_n,\,x_n^{-1}\rangle$. Under what ...
3
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2answers
92 views

Semilocal commutative ring with two or three maximal ideals [closed]

Is there any equivalence condition for a commutative ring to have exactly two or three maximal ideals?
5
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1answer
52 views

Is the ring of Laurent polynomials in $n$ noncommuting variables Noetherian?

Suppose we have a Noetherian ring $R$. Is it true that the ring of Laurent polynomials $R\langle x_1,\,x_1^{-1},\ldots,\,x_n,\,x_n^{-1}\rangle$ in $n$ noncommuting variables is also Noetherian? If so, ...
0
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3answers
47 views

How to determine non trivial homomorphisms [closed]

I am trying to understand and it doesn't make any sense to me: How can I determine if there are any non trivial homomorphisms between groups or rings? How do I find them? and once I found them, how ...
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0answers
25 views

Noether & Schmeidler- Hurwitz-Ideals

Consider the following page from Noether and Schmeidler's 1921 work: http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN266833020_0008&DMDID=DMDLOG_0008&LOGID=LOG_0008&PHYSID=PHYS_0013 ...
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2answers
74 views

Dimension of the affine variety associated to $\langle zw-y^2, xy-z^3 \rangle $

Find the dimension of the affine variety $V(I)$, where $I=\left\langle zw-y^2,xy-z^3\right\rangle \subseteq k[x,y,z,w]$, with $k$ algebraicaly closed field. I tried to solve the system $zw-y^2=0$, ...
2
votes
1answer
58 views

What does $(G:G')$ mean?

I'm trying to teach myself some ring theory from a book, and have come across this sentence: "There are $(G:G') = 4$ linear characters" where $G$ is a group, and $G'$ is the derived group. I ...
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2answers
62 views

Prove that field $Q(x)$ is a field of fractions of ring $F[x]$

Let $F$ be a commutative ring without zero divisors and $Q$ its field of fractions. How can I prove that field $Q(x)$ is a field of fractions of ring $F[x]$? And also why is it that field $Q((x))$ ...
2
votes
0answers
114 views

Left Ideals and Two-Sided Ideals of Rings of Matrices

This is a question I posted in this topic: Pathologies in "rng". However, I decided that the question deserves its own thread, and I want to know if anybody can answer it. For ...
2
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1answer
37 views

Show that $G$ is a basis of group ring $RG$ over $R$.

Show that $G$ is a basis of group ring $RG$ over $R$. Comments: That $G$ is through direct generator, if $\alpha \in RG$ then $\alpha = \sum_{g \in G} a_gg$. I am not able to show that $RG$ is a ...
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0answers
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Smallest Two-Sided Nearring

For those who are unfamilar with nearrings, here is a definition. A two-sided nearring is a triplet $(N,+,\cdot)$, where $(N,+)$ is a group and $(N,\cdot)$ is a semigroup such that we have both left ...
0
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1answer
46 views

Please give an example of a ring that does NOT have a multiplicative identity but contains a subring that does have an identity..

I cannot think of an example of such a ring. For that matter, other than the even integers I cannot even think of an example of a ring without an identity.
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5answers
818 views

Pathologies in “rng”

There is no general consensus regarding whether a ring should have a unity element or not. Many authors work with unital rings , and other does not essentially require unity. If we do not assume ...
5
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0answers
24 views

Why not take the tensor product of two left modules in this way? [duplicate]

Let $A,B$ be two left $R$-modules. I was wondering if we then can form the tensor product of $A$ and $B$ by the free abelian group on $A \times B$ divided out by the span of the following elements ...
1
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1answer
42 views

How does one find the Krull dimension of a composite ring?

For example, if the ring is $\mathbb{Z} + X \mathbb{Q}[X]$. Is the dimension $1$?
1
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1answer
33 views

Why is $(2,x)$ non-principal in $\Bbb Z[x]$? [duplicate]

Why is $(2,x)$ non-principal in $\Bbb Z[x]$? Apparently this is the case, I just read it on wiki, as a counter example to $\Bbb Z[x]$ being a PID. What is $x$ here? I mean $2$ can surely generate ...
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0answers
40 views

Ring structure of a localization

let $R$ be a commutative noetherian ring and let $A$ be an $R$-algebra which is moreover a finitely generated $R$-module. Let $P$ be a prime ideal of $R$. How is the ring structure of the localization ...
0
votes
1answer
34 views

Finding all homomorphisms between rings

I am looking for a good method to understand how to find all possible homomorphisms between rings, e.g $\varphi :\mathbb{Z}\rightarrow \mathbb{Z}$ or, as another example: $\varphi ...
1
vote
1answer
42 views

Find all subrings of a ring

Given a finite ring, e.g $\mathbb{Z}{_{24}}$, how can I find all of its subrings? I have tried to think about it couldn't reach any idea. Thanks.
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1answer
83 views

About the usage of the strong approximation theorem

I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says: Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is ...
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1answer
14 views

Prove $\operatorname{ann}_r(S)$ is an ideal.

Let $S$ be a right ideal of ring $R$. Let $\operatorname{ann}_r(S) = \{r \in R:ar=0 \mbox{ for all } a \in S\}$. Prove $\operatorname{ann}_r(S)$ is an ideal of $R$. I have practically no experience ...
4
votes
2answers
275 views

Example: Krull dimension 1 but not a PID

It's easy to prove that if $A$ is a PID which is not a field then $\dim A= 1$. What is a counterexample to the converse? Thanks for any insight.
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2answers
28 views

Invertibility of translating ideals from a ring to its localization

Let $A$ be a ring with some multiplicative subset $S$. Define $AS^{-1} = \{\frac as| a \in A, s \in S\}$. Let $I_A$ and $I_S$ be the sets of ideals of $A \subset A-S$ and $AS^{-1}$ respectively. ...
0
votes
1answer
69 views

Finding the quotient ring $\mathbb{Z}[i]/(4+i)$

Find the quotient ring $\mathbb{Z}[i]/(4+i)$ by identifying elements with the lattice points in the square generated by $4+i$. I know that $N(4+i) = 17$. Therefore, $4+i$ is irreducible. Now ...
2
votes
1answer
30 views

Computing injective hulls over a lower triangular matrix ring

Let $R$ be the ring $\begin{pmatrix} {\mathbb Z }_{ p } & 0 \\ {\mathbb Z }_{ p } & {\mathbb Z }_{ p } \end{pmatrix}$ with $p$ prime. $R$ is an artinian ring and is also a $\mathbb ...
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votes
3answers
47 views

Proof of ring isomorphism

Proof that $Z[X]/(X^2-22) ≈ Z[\sqrt{22}]$. I have tried all sorts of things to resolve this but I don't know how to wrap my head around it. Can you please explain how to solve these kind of ...
0
votes
1answer
22 views

$E_{i,j} = E_{i,r} * E_{r,s} * E_{s,j} (i,j=1,2,…,n)$ where $E_{x,y}$ is an $n \times n$ matrix with a 1 in row x column y and 0's otherwise.

This matrix equation was given in my ring theory text as a step in showing that there are no nontrivial proper ideals of the ring of $n \times n$ matrices with real number entries. I am assuming it ...
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2answers
47 views

Are these rings fields?

Are the following rings fields? 1) $\Bbb Q[x] /\langle x^2+1\rangle$ Since a polynomial ring taking values on any field is a E.D, and hence a P.I.D, this is a field iff the ideal is prime or ...
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1answer
29 views

Quotient ring is a ring homomorphism

Why is this a ring homomorphism? $$\phi:R\to R/I$$ where $I$ is an ideal, given by $\phi:r\mapsto r+I$. To be a ring homomorphism it needs to be a homomorphism of addition and multiplication, i.e: ...
3
votes
1answer
52 views

Are these subrings of $\Bbb Q$?

Are the following subrings of $\Bbb Q$? 1) The set of non-negative rational numbers. No since we don't have any additive inverses, and the subring should be armed with an Abelian group for ...
0
votes
3answers
32 views

Prove for every $a$ in $I$ and every $b$ in $J$ that $ab=0$.

$I$ and $J$ are respectively right and left ideals of ring $R$. $I$ and $J$ have no elements in common other than $0$. Prove for every $a$ in $I$ and every $b$ in $J$ that $ab=0$. I have ...
2
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2answers
44 views

Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring)

Prove that: $B/A \triangleleft R/A$ if $A \subseteq B ;\ \ A, B \triangleleft R $(ring) : $ \triangleleft $ means ideal. I need this proof to continue on, I'm told it's not that hard, but I just ...
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votes
0answers
49 views

Determine all maximal and prime ideals of the polynomial ring $\Bbb C[x]$

Determine all maximal and prime ideals of the polynomial ring $\Bbb C[x]$ My attempt: Note that $\Bbb F[x]$ where $\Bbb F$ is any field is a Euclidean domain, and importantly, that means that ...
4
votes
3answers
33 views

Is the set of all rational numbers with odd denominators a subring of $\Bbb Q$?

Is the set of all rational numbers with odd denominators a subring of $\Bbb Q$?(When the fraction is completely reduced) I have tried to apply the subring test on this, and this means I want to show ...