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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What is the difference between submodules of $A/\mathfrak a$ as an $A$-module or as an $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, Then we can see to $A/\mathfrak a$ as an $A$ module or as an $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structure difference ...
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24 views

Show that every short exact sequence $0\rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ (with $M''$ free) is split exact

Note: $M,M',M''$ are modules. In order to show that it's split exact, I understand that I need to show that $\beta:M\rightarrow M''$ has a left inverse, and similarly that $\alpha:M'\rightarrow M$ ...
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0answers
11 views

recover (pontrjagin) ring structure from the localization (w.r.t. $\pi_0$)

Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect ...
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15 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
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3answers
298 views

Example of a ring with infinitely many zero divisors and finitely many invertible elements

I am preparing to my abstract algebra exam and I try to find an example of a ring with infinitely many zero divisors and finitely many invertible elements (rather simple if possible). Does it even ...
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1answer
21 views

Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$

Let $F$ be a commutative ring without zero divisors and $Q$ -its field of fractions. Let $Q(x)$ be also field of fractions of ring $F[x]$. How can field $Q((x))$ not match field of fractions of ring ...
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4answers
379 views

Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...
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1answer
33 views

Is there a way to generate groups, rings, fields, etc.? [on hold]

There are ways to generate a list of numbers $a_1, a_2,...a,n$ such that no $a_i,a_j$ share any factors, mainly by letting $a_{n+1}=a_n^2-a_n+1$ with $a_0>1$, my question is: is there a way to ...
3
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1answer
104 views

What motivates the definition of a ring in abstract algebra? [on hold]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for ...
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2answers
32 views

There is no nontrivial ring homomorphism between two commutative rings with unity and characteristic of distinct primes

The following is an old exam question and the question is: Show that there is no nontrivial ring homomorphism between two commutative rings with identity if their characteristics are distinct primes. ...
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1answer
64 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
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2answers
70 views

The subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$ is not finitely generated over $k$ [duplicate]

Let $R$ be the subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$. Can someone explain why $R$ is not finitely generated as a ring over $k$ (i.e. finitely generated as a $k$-algebra)? By ...
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1answer
39 views

Prime ideals in $R[x]$, $R$ a PID

Let $R$ be a PID. Show that if $r \in R$ and $$p = (r, \underline{f}(x), \underline{g}(x))$$ is prime, where $\underline{f}(x), \underline{g}(x) \in R[x]$ are nonconstant irreducible polynomials, ...
5
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2answers
74 views

Prime ideal $P$ of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z}=\{0\}$ is principal

The problem stated more precisely is this: Let $P$ be a prime ideal of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z} =\{0\}$. Show that $P$ is a principal ideal. I think there is a problem with my ...
3
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1answer
30 views

The quotient of a direct sum of rings

Is the quotient of a direct sum of rings isomorphic to the direct sum of the quotients? $$ (R_1 \oplus R_2 ) / \langle (x_1, x_2)\rangle = (R_1 / \langle x_1\rangle ) \oplus (R_2 /\langle x_2\rangle) ...
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2answers
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The ideal generated by the set $I \cup \{a\}$.

Let $I$ be an ideal of $R$ a commutative ring with identity. For some $a \in R$, prove that the ideal generated by $I \cup \{a\}$, denoted $(I,a) = \{ i + ra \mid i \in I \text{ and } r \in R\}$. My ...
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1answer
25 views

Example of a Non-Graded Ideal in a Graded Ring

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$. An ideal $I$ in a ...
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4answers
49 views

If $R$ is a ring with identity and $a$ is a unit, prove that the equation $ax=b$ has a unique solution in $R$.

So, this was my initial proof: Assume $R$ is a ring, and $a,b\in R$ Let $x_1$ and $x_2$ be solutions of $ax=b$ Hence, $ax_1=b=ax_2 \Rightarrow ax_1-ax_2=0_R \Rightarrow a(x_1-x_2)=0_R$ Thus, we ...
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1answer
113 views

Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s

Motivated by this question, I want to find a complete set of relations for the ring of integer-valued polynomials, where the generators are the polynomials $\binom{x}{n}$ for $n\in \mathbb{N}$. The ...
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1answer
18 views

What is a relation (finitely related module)?

https://en.wikipedia.org/wiki/Finitely-generated_module#Finitely_presented.2C_finitely_related.2C_and_coherent_modules I've understood the first part of the definition. Then, "M is isomorphic to ...
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2answers
49 views

Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
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1answer
13 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 ...
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1answer
57 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 ...
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4answers
39 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
33 views

Prime ideal in indecomposable commutative ring [on hold]

Let $R$ be a commutative indecomposable ring with Jacobson radical $J$. When can we find a prime ideal contained in $J$?
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27 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 ...
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1answer
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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 ...
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1answer
41 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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26 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
55 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 ...
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2answers
40 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
29 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
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$\mathbb Z_n$ 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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2answers
50 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 ...
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1answer
25 views

Radical of An Ideal In $Z_n$

I am searching for an answer in $Z_n$ regarding the radical of an ideal. Consider an ideal $(a)$ in $Z_n$. Can we calculate radical of $(a)$ in general?
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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 $\Bbb{Z}[\sqrt{D}] = \{a + b \sqrt{D} | a,b, \in \Bbb{Z}\}$ ...
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1answer
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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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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 $$ ...
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1answer
78 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) ...
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1answer
69 views

Does there exist any ring $R$ such that $R^n\cong R^m$ for all $n,m\in\mathbb{N}$? [duplicate]

Does there exist any ring $R$ such that $R^n\cong R^m$ for all $n,m\in\mathbb{N}$? Or for some $n,m\in\mathbb{N}$ such that $m\ne n$?
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1answer
10 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 ...
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2answers
76 views

Semilocal commutative ring with two or three maximal ideals

Is there any equivalence condition for a commutative ring to have exactly two or three maximal ideals?
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1answer
48 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, ...
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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
29 views

searching about an ismorphism

I'm looking for an isomorphism : $$H: \overbrace{\mathbb{F}_q^r\oplus\ldots\oplus \mathbb{F}_q^r}^{l\ times}\longrightarrow \frac{\mathbb{F}_q[X_1,\ldots,X_l]}{(X_1^r-1)\ldots(X_l^r-1)}$$ ??
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23 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
65 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
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1answer
56 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
57 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 $Q((x))$ ...
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0answers
31 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 ...