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
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Extensions between integral domains give extensions of fields of the same degree.

Assume that $S \subset R$ is a ring extension where, both $S$, $R$ are integral domains. Furthermore, assume that $R$ is a free $S$-module of rank $n$. Is it true that the extension of fields $\mathrm{...
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0answers
53 views

Clifford algebra of a non-diagonal quadratic form over rings

I know how to construct explicitly the clifford algebra of a quadratic form over fields, even in the case the diagonal quadratic form over rings. But how should I $\textbf{construct explicitly}$ the ...
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4answers
66 views

Showing that two quotient rings are isomorphic

Is $\mathbb{Q}[x]/(x^2-x-1)$ isomorphic to $\mathbb{Q}[x]/(x^2-5)?$ My guess is yes. I am trying to find an isomorphism between the two. Universal Property of Quotient certainly helps. I am thinking ...
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0answers
32 views

does algebra over algebraically closed field has isomorphism classes of irreducible modules?

Let $F$ be an algebraically closed field and $M$ be an F-algebra. Which are the conditions that make $M$ has finitely many isomorphism classes of irreducible $M$-modules?
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1answer
30 views

What does “coefficients from all of $\mathbf{F} _q$” mean

I was reading Wikipedia's page on Ring Learning with Errors, and came to wonder what is meant by "with coefficients from all of $\mathbf{F} _q$" which is a requirement for the set of known polynomials....
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1answer
46 views

Atiyah–Macdonald exercise 14 chapter 1

So here is the part of exercise 14 of chapter 1 that has been bothering me: Let $A$ be a commutaive ring with identity. Let $\Sigma $ be the set of ideals with the property that every element in them ...
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0answers
56 views
+100

Does every finite dimensional real nil algebra admit a multiplicative basis?

We say that a finite dimensional real commutative and associative algebra $\mathcal{A}$ is nil if every element $a \in \mathcal{A}$ is nilpotent. By multiplicative basis, I mean a basis $\{ v_1, \...
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1answer
56 views

If $B$ is an ideal of $A$ then $B[x]$ is an ideal of $A[x]$ - what's wrong with my proof?

This is exercise E.2 from chapter 24 of Pinter's A Book of Abstract Algebra: If $B$ is an ideal of $A$, $B[x]$ is not necessarily an ideal of $A[x]$. Give an example to prove this contention. It ...
2
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1answer
35 views

Mal'cev condition for variety of rings generated by finite fields to be arithmetical

This is an exercise of Burris & Sankappanavar (Universal Algebra), Chapter II, section 12. It asks to prove that, if $V$ is a variety of rings generated by finitely many finite fields, then $V$ is ...
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0answers
26 views

Minimal prime ideals of the ring of continuous functions

Let $X$ be a topological space. Are there any conditions on $X$ which guarantee that that the minimal prime ideals of $C(X)$, the ring of real-valued continuous functions on $X$, have a nice ...
1
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1answer
47 views

Ring->module->$R$-algebra, Field->Vectorspace->algebra

I haven't done any mathematics for a long time, and I have forgotten some things. I want to try to remember some of the words and how they interact. A module is a 'vectorspace over a ring' rather ...
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0answers
15 views

Finitely generated module: terminology.

What's the meaning of the expression: $S$ is a subring of $\mathbb{C}$ finitely generated as $\mathbb{Z}$-module? Maybe that the additive group of the ring $S$ is a finitely generated abelian group? ...
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2answers
37 views

assuring factorization for R[x] when R is a UFD

I wanted to ask, suppose the ring $R$ is a UFD (Unique factorization domain) and I look at $R[x]$, the ring of polynomials over $R$. I wanted to know, how can I assure that when I have some polynomial ...
2
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1answer
36 views

If the ring $M_n(D)$ is a $k$-algebra, is $D$ a $k$-algebra?

Let $k$ be a field, let $D$ be a division ring. Assume the matrix ring $A = M_n(D)$ is endowed with some $k$-algebra structure compatible with its ring structure (namely, for all $\lambda \in k$ and ...
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0answers
23 views

$F(X)$ as a subfield of $F((X))$ of formal Laurent series

$F(X)$ is a subfield of $F((X))$ by considering the Laurent expansion of rational functions at the origin. So what is indeed the degree of this field extension $F((X))/F(X)$? Or this is an infinite ...
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2answers
34 views

How do I see that $a = bu$ for some unit $u$? [duplicate]

Suppose elements $a$ and $b$ in a domain satisfy $a \mid b$ and $b \mid a$. How do I see that $a = bu$ for some unit $u$?
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2answers
44 views

Examples of $R$-modules $X$ such that $(X \setminus TX) \cup \{0\}$ isn't a submodule.

Work over an ambient commutative ring with unity. Given a module $X$, write $TX$ for its submodule of torsion elements. Suppose we want to find the "submodule" of torsion-free elements of $X$. So ...
2
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1answer
39 views

$a \in \mathbb{Z}[i]$ is a unit if and only if $a$ divides every element of $\mathbb{Z}[i]$? [closed]

As the question title suggests, how do I see that $a \in \mathbb{Z}[i]$ is a unit if and only if $a$ divides every element of $\mathbb{Z}[i]$?
3
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1answer
27 views

Does it necessarily follow that $a = ub$ for some unit $u \in \{\pm1, \pm i\}$? [closed]

Suppose that $a$, $b \in \mathbb{Z}[i]$ satisfy $a \mid b$ and $b \mid a$. Does it necessarily follow that $a = ub$ for some unit $u \in \{\pm1, \pm i\}$?
2
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1answer
50 views

A doubt about the correspondence theorem.

Let $f$ be a ring homomorphism from $R$ onto $R_1$. Then there is a one one correspondence between the set of all ideals of $R_1$ and the set of all ideals of $R$ that contain the kernel. Now what ...
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3answers
55 views

$\ker \phi = (a_1, …, a_n)$ for a ring homomorphism $\phi: R[x_1, …, x_n] \to R$

Let $R$ be a commutative ring, $a_1, ..., a_n$ its elements and $\phi: R[x_1, ..., x_n] \to R$ defined by $ \phi(f(x_1, ..., x_n)) = f(a_1, ... ,a_n)$ a ring homomorphism. Prove: $\ker \phi = (...
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2answers
25 views

Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$

Let $\phi: R \to S$ be a surjective homomorphism. Prove that $\frac{R/ \ker \phi}{(\ker \phi + J)/ \ker \phi} \cong \frac{S}{\phi(J)}$ for an ideal $J$ of $R.$ Obviously, $S \cong R/ \ker \phi$(first ...
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0answers
48 views

Let $k$ be a field of characteristic $\neq 2$. Then $x^6-xy+y^6$ is irreducible in $k[x,y]$. [duplicate]

There is no obvious way to apply Eisenstein's criterion; and if I assume by contradiction that $x^6-xy+y^6=f(x,y)g(x,y)$, f with homogeneous degree $\leq 3$. Then I have that $f(x,y)=\sum_{i,j} a_{ij}...
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0answers
38 views

Idempotents central or not?

Let $R$ be a nil-clean ring with unity such that $R/J(R)$ is reduced, where $J(R)$ is the Jacobson radical of $R$. Is it true that $R$ is abelian, i.e. the idempotents are central? (By nil-clean I ...
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0answers
23 views

How does restriction of scalars interact with tensor products?

Say that we have a morphism of commutative rings $f: R \to S$. Does the restriction of scalars functor $f^*: S \text{Mod} \to R \text{Mod}$ commute with tensor products? In other words, I would like ...
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2answers
54 views

Intuitive reasons of ring modulo maximal ideal or prime ideal

Are there any intuitive reasons that can help us remember that $R/I$ is a field iff $I$ is a maximal ideal; $R/I$ is an integral domain iff $I$ is a prime ideal? (I can understand the proof, but have ...
0
votes
1answer
63 views

What do we mean by a ring and what is ringlike about it? [duplicate]

I see that a ring is a triple $(R,\cdot,+)$. I am confused by the terms abelian group and semigroup. Does this mean for $x \in R$ and $y \in R$, $x \cdot y$ and $x+y$ are defined? If so, how is this ...
2
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2answers
54 views

Notation for the set of zero divisors in a ring

If $R$ is a nonzero ring with identity then I have seen the group of units denoted by $R^{\times}$ or possibly $R^*$ in some texts. In a classical ring there is a trichotomy which declares each ...
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0answers
35 views

Dimension under integral local homomorphism

Let $f:(R,m) \to (S,n)$ be an integral local homomorphism. Let $p$ be a prime ideal of $R$ not equal to $m$. I want to know if one can claim $\dim S/f(p)S\neq0 $. This is true when $(f(p)S)^c=p$, ...
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0answers
39 views

A monomorphism from an $R$-module to $R$

Let $R$ be a commutative ring with unity possessing an element $r$ in the singular ideal $Z(R)=$ the set of elements whose annihilators are essential in the module $R_R$, and let $M$ be a faithful $R$...
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0answers
24 views

Meaning of “scaled topology” on a ring

Suppose that $(A,\tau)$ is a DVR with a topology (which is not the metric topology) with field of fractions $K$. In a conference the speaker said that for any $b\in K^\times$ one can define the "...
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1answer
24 views

Counit for the restriction of scalars, extension of scalars adjunction

Let $f: R \to S$ be a morphism of noncommutative rings. Let $$f_!:= S \otimes_R (-) : R \text{Mod} \to S \text{Mod}$$ denote the extension of scalars functor, and $$f^*: S \text{Mod} \to R \text{Mod}$...
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0answers
44 views

Irreducibility of sums of two polynomials

I'm interested in a special type of polynomial factorization over $\mathbb {Q} $: testing the irreducibility of $f(x)+g(x)$, where $f$ and $g $ are relatively prime and $\text {deg}(f)<\text{deg}(g)...
3
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0answers
71 views

Can we continually factorize an expression like $x+y$?

I have a question that, for lack of familiarity or understanding of the relevant fields, I'm not quite sure how to formulate, so I'll just start off with an example and list some questions as I go. As ...
3
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2answers
37 views

$R/(IJ)$ is reduced $\Rightarrow IJ = I \cap J$ for ideals $I,J$ of a commutative ring $R$

This is exercise $4.6$ on page $154$ of the textbook Algebra: Chapter $0$ (authored by P. Aluffi): Let $I,J$ be ideals of a commutative ring $R$. Assume that $R/(IJ)$ is reduced (that is, it has ...
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2answers
38 views

Generalizing concept of content of a polynomial to commutative rings [duplicate]

Let $A $ be a commutative ring with identity. Let $f,g\in A [x] $. Let $I_1,I_2, J $ be the ideals generated by the coefficients of $f,g,fg $ respectively. Must $J $ be equal to $I_1 I_2$ ? It is an ...
3
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2answers
102 views

In abstract algebra, what is an intuitive explanation for a field?

Wikipedia has the following to say about fields. In mathematics, a field is one of the fundamental algebraic structures used in abstract algebra. It is a nonzero commutative division ring, or ...
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3answers
351 views

A few questions on the Gaussian integers

I have a few questions surrounding the Gaussian integers, which I hope can be answered together in one fell swoop. The Gaussian integers are defined as $\mathbb{Z}[i] = \{x + iy : x, y \in \mathbb{Z}...
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1answer
43 views

Intuition for basic fact surrounding Gaussian integers.

What is the intuition behind the following fact? Among the odd primes: Those that have remainder $3$ upon division by $4$ remain prime in $\mathbb{Z}[i]$. Those that leave remainder $1$ ...
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1answer
34 views

Is there a name for rings of the form $R/\mathfrak{q}$ with $\mathfrak{q}$ primary?

In a unital commutative ring $R$, a lot of the times we characterize the property of an ideal $\mathfrak{a}$ by the property of the quotient ring $R/\mathfrak{a}$. Examples are if $\mathfrak{a}$ is ...
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0answers
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How do you create a Gaussian distribution on a polynomial ring?

In the specification for the YASHE homomorphic encryption algorithm, it says that for the parameter generation subroutine, you need to: Given the security parameter $λ$, fix ... distributions $\...
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0answers
36 views

Polynomial ring operations on $\mathbb{Z}$

The usual ring operations on $\mathbb{Z}$ can be defined via polynomials in $\mathbb{Z}[a,b]$ (when viewing $a,b$ as variables): Addition: $(a,b) \mapsto a+b \in \mathbb{Z}[a,b]$ Multiplication: $(...
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2answers
39 views

name of a certain class of rings

Does there exist a name for the class of commutative rings with identity that satisfy the following: For any 2 ideals $I_1,I_2$ of R,we have : $I_1 I_2= (I_1\cap I_2)(I_1+I_2) $ I would also like to ...
1
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1answer
32 views

Every ideal is the sum of principal ideals implies ascending chain of ideals is finite?

I am looking at this problem at the moment: If R is a commutative ring with 1, in which every ideal is the sum of finitely many principal ideals $(I=r_1R+r_2R+...+r_nR)$ , show that this implies, ...
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0answers
49 views

What is the relation between the two following dimensions in polynomial ring?

Let ‎$‎I‎$ ‎be an ‎ideal ‎of $‎R=\Bbb K[x_1,‎\ldots‎,x_n]‎$‎‎ and ‎$‎{\bf u}=\{x_j\}_{j=1}^{\ell<n}‎\subset‎\{x_1,‎\ldots‎,x_n\}‎$‎‎‎. $‎{\bf u}‎$ ‎is called ‎independent ‎modulo ‎‎$‎I‎$ ‎if ‎‎$I\...
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2answers
51 views

Cardinality of the base of a ring of sets

Concretely, my question is: What is the size of the minimal base of a ring of sets? In my understanding the base is the set of elements that can be used to "deduce the rest". E.g., if I have a ring ...
3
votes
1answer
46 views

Is $x^3+y^3+z^3-1$ irreducible in a field $k$ of characteristic $\neq 3$?

If I can show that $y^3+z^3-1$ is irreducible, I believe I can use Eisenstein's criterion. But I don't see how to show even this. The other approach on my mind is to show that $k[x,y,z]/(x^3+y^3+z^3-...
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2answers
46 views

There is only one structure of ring(with identity) on abelian group $(\mathbb{Z},+)$. Prove that a certain ring homomorphism is surjective.

This is an exercise from a textbook "Algebra: Chapter 0" by Paolo Aluffi. First, I state necessary facts from the book used in the exercise: For every abelian group $G, End_{Ab}(G)$( the set of ...
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0answers
25 views

quotient algebra and left module structure

Pardon me if the question might be obvious: Let $L/A$ be a quotient algebra (associative) then we want to show that $L/A$ is a left $A$-module?
1
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2answers
103 views

Nilpotent or non-Nilpotent Jacobson Radical

Let $R$ be a ring with identity element such that every ideal of which is idempotent or nilpotent. Is it true that the Jacobson radical $J(R)$ of $R$ is nilpotent? If $R$ is Noetherian and $J(R)$ is ...