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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2answers
66 views

Can $R \times R$ be isomorphic to $R$ as rings?

I know from this question that a ring $R \times R$ can be isomorphic to $R$, as $R$-modules. But can they ever be ismorphic as rings?
3
votes
2answers
38 views

Kahler differentials and quotient rings.

I am dealing with some nice rings that are always isomorphic to some fairly nice quotient ring of a polynomial ring. A typical example is: $$ \mathbb{C}[X,XY,XY^2] \cong ...
1
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1answer
38 views

homomorphic maps between the rings $\mathbb{Z}/12\mathbb{Z}$ and $\mathbb{Z}/42\mathbb{Z}$.

I wanted to find all the homomorphisms $\theta : \mathbb{Z}/12\mathbb{Z} \ \rightarrow \ \mathbb{Z}/42\mathbb{Z} $. I thought that it would be enough to describe the map by $1 \mapsto a$ for some ...
0
votes
0answers
45 views

Non-existance of a neutral element for the cup product

I know that if $X$ is a space and $R$ a commutative unitary ring, the cup product $$\smile : H^k(X; R) \times H^l(X;R) \rightarrow H^{k+l}(X;R)$$ has as a neutral element the cohomology class in ...
2
votes
2answers
51 views

Can units in $M_n(\mathbb{Z})$ be moved to the other side?

Let $M, U_1 \in M_n(\mathbb{Z})$ with $U_1$ a unit (i.e. $\lvert \det(U_1) \rvert=1$). Can I always find another unit $U_2\in M_n(\mathbb{Z})$ such that $U_1 M = M U_2$?
1
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0answers
40 views

Is this a fruitful enrichment of $R[X]$?

Let $R$ be a commutative ring. Then polynomial ring $R\left[X\right]$ can be looked at as an $R$-algebra free over a singleton. If $S$ is another $R$-algebra then for any element $s\in S$ there is a ...
2
votes
3answers
109 views

What's $R[x]/(x^2)$ isomorphic to?

Can someone explain what the quotient ring $R[x]/(x^2)$ is isomorphic to? I know it's weird because it's reducible/has double root, but I'm not exactly sure what the implications of that, or how to ...
3
votes
0answers
45 views

Find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$.

Let k be any field, then how does one find valuation rings of the function field $k(x,y)/k(xy)$ which do not contain $k[x,y]$? I believe there are two. If we consider a rational function field ...
1
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0answers
36 views

Depth and kernel of multiplication map

If $A$ is a commutative $k$-algebra, $\mu: A\otimes_k A\rightarrow A$ is its multplication map and $I$ is the kernel of that map (viewed as an $A\otimes_k A$-module map) then what is the relationship ...
1
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0answers
50 views

Krull dimension of localization

If $R$ is a commutative ring and $m$ a maximal ideal therein, then what are the conditions for the Krull dimension of $R$ equaling to the Krull dimension of $R_m$?
0
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0answers
31 views

Looking for an example of a norm's kernel that is not an ideal

In order to formulate the question we need a couple of notions. (All rings mentioned are unital and nonzero, all modules are unitary.) Let $k$ be a ring, $A$ a $k$-module, and $f\colon A\to S$ a ...
0
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0answers
17 views

Is there any semiring such that every element $z$ only has sum decomposition of $0+z$ and some another decomposition?

Let us say that there is a semiring $R$. By properties of semi-ring, every element $x \in R$ is equal to $0+x$. Is there any nontrivial semi-ring that every element $x \in R$ has only one other finite ...
1
vote
3answers
124 views

What is so special about $a*b^{ -1}$ equivalence?

This equivalent is used often in group theory. For example, using this equivalnce you prove Lagranges theroem and also this equivalence gives you the cosets and other things. This equivalence also ...
4
votes
1answer
121 views

Free modules basic understanding problem

I have been told that the $\mathbb{Z}$ module $\mathbb{Z}/2\mathbb{Z}=\{0,1\}$ isn't free. For a module to be free, there must exist a subset such that every element is expressible as a finite linear ...
2
votes
1answer
43 views

An easy looking quotient of a local ring

$k$ is a number field, $R$ its ring of integers and $\mathfrak p$ a nonzero prime ideal of $R$. Let $R_\mathfrak p$ be the localization of $R$ at $\mathfrak p$. Is it true that $R_\mathfrak ...
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3answers
100 views
+50

Some practical questions on cohomology and the ring $\mathbf{Z}[x]/(x^2)$

So I know that the cohomology ring of $S^n$ is $\mathbf{Z}[x]/(x^2)$ with "$x$ in degree $n$"; if we ignore the grading then this ring fails to distinguish the spheres. What is actually meant by "in ...
10
votes
1answer
92 views

Could one make a ring of matrices of uncountable size?

I've seen several kinds of matrices. You could see a real square matrix as a mapping: $$ A \quad : \quad \{1, 2,\cdots, n \}^2 \ \longrightarrow \ \mathbb{R} $$ I've seen that there were also infinite ...
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3answers
47 views

The number of ideals in the quotient ring $\mathbb R[x]/\langle x^2-3x+2 \rangle$ [duplicate]

Finding the number of ideals in the quotient ring $\mathbb R[x]/\langle x^2-3x+2 \rangle$. Attempt: $R[x]/\langle x^2-3x+2 \rangle = \{f(x)+\langle x^2-3x+2 \rangle~~|~~f(x) \in R[x]\}$. Since ...
1
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1answer
16 views

Irreducibles - difficulty with the definition

I'm working from the definition that in an integral domain $R$, an irreducible is an element $p$ such that if $p=xy$ then either $x$ or $y$ is a unit. In certain proofs on my course, the lecturer has ...
1
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1answer
28 views

Unqiue Factorization Domains, is the product finite?

Having looked around a bit, the most common definition of a UFD is an integral domain such that any element can be expressed as a product of a unit and irreducible elements, and that this ...
1
vote
1answer
29 views

Suppose that $\beta$ is a zero of $f(x)=x^4+x+1$ in some field extensions of $E$ of $Z_2$.Write $f(x)$ as a product of linear factors in $E[x]$

Suppose that $\beta$ is a zero of $f(x)=x^4+x+1$ in some field extensions of $E$ of $Z_2$.Write $f(x)$ as a product of linear factors in $E[x]$ Attempt: In $\mathbb Z_2: \beta^4+\beta+1=0$ Going by ...
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3answers
39 views

why is maximal ideal of a ring of integers generated by a single prime number?

I cannot understand why maximal ideal in a ring of integers is generated by a single prime number. For example, if we choose 2 and 3 as generators of ideal, then all multiples of 2 and 3 will be in ...
0
votes
1answer
68 views

Canonical ring map

Let $\chi:\mathbf{Z}\rightarrow A$ be the canonical map to a ring $A$, and let $p$ be a prime ideal of $A$. Then I claim that $\chi^{-1}(p)=(\mathrm{char} \ k(p))$ where $k(p)$ is the residue field at ...
1
vote
2answers
30 views

Some irreducible polynomial

Is the polynomial given by $y^2-p(x)\in C[x,y]$ with $p$, all of whose roots are distinct, an irreducible polynomial? Interesting is when $p$ has degree $3$ innit
0
votes
1answer
38 views

$R$ is a ring. $(R,+)\cong(Z_2\oplus Z_4,+)$, indecomposable, $\nexists$ 1, noncommutative, has an idempotent $e\neq 0$. Show that $2e\neq 0$.

Let $R$ be an indecomposable ring with $(R,+)\cong(\Bbb{Z}_2\oplus \Bbb{Z}_4,+)$. Suppose that $R$ is noncommutative and has no multiplicative identity. If $R$ has an nonzero idempotent element $e$, ...
1
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2answers
41 views

Finitely generated ideal with special property

Is there a ring with a finitely generated ideal $I$ which has an infinite subset $M\subseteq I$ such that $M$ generates $I$ but no finite subset of $M$ does it? What I found out: If such a rings ...
-1
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0answers
22 views

A commutative ring that satisfies $(x=0) \rightarrow ((\forall z \forall y \,\, x = yzz) \vee (\forall y \forall z \,\, x=yz+zy))$

Can anyone give an example of a commutative ring that satisfies $(x=0) \rightarrow ((\forall z \forall y \,\, x = yzz) \vee (\forall y \forall z \,\, x=yz+zy))$ and $\forall x \,\, x \cdot x = 0$? ...
1
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1answer
51 views

Question concerning a list sorting problem

I have the following question: Let $a,b,c,d$ be four natural numbers with $a \leq b$ and $c\leq d$. I have written a program that produces a list, which has as entries all 2-tuples $(x,y)$ with ...
1
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0answers
50 views

Direct proof that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p^2\mathbb{Z}$-module.

I am trying to prove that $\mathbb{Z}/p\mathbb{Z}$ is not a flat $\mathbb{Z}/p\mathbb{Z}$-module. The reasoning I have is the following. We have an exact sequence $0 \to \mathbb{Z}/p\mathbb{Z} ...
4
votes
2answers
103 views

Commutative and anticommutative ring that respects $x^2 = 0$, where $0$ is additive identity

Can anyone present an example where a ring consists of infinitely many different sets/elements in a universe (ring), is both commutative and anticommutative, and respects $x \cdot x=x^2 =0$ for ...
0
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0answers
11 views

How to create a commutative lie algebra from commutative ring?

How does one create a commutative lie algebra (lie algebra is inherently anti-commutative, so this is added restriction) from commutative ring?
5
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0answers
129 views

Showing $\sqrt{2} \notin E.$

Let $E$ denote the least subring of $\mathbb{R}$ that is closed under the operation $r \mapsto e^r$. Then presumably, $\sqrt{2} \notin E.$ Question. How can we show this?
0
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1answer
18 views

Finitely generated quotient module isomorphism

In an assignment, I was asked to prove that if $M$ is an $R$ module and $N\subseteq M$ is a submodule such that $M/N$ is finitely generated and free, then there exists a submodule $F$ of $M$ which is ...
2
votes
1answer
23 views

More classes of rings spanned by units?

I will formulate questions only after a rather protracted introduction. Rings in this text are unital and nontrivial, algebras are associative, modules are unitary. ...
1
vote
2answers
52 views

Calculating the reminder when dividing complex numbers

When dividing complex numbers, let us consider $\frac{26+120 i}{37+226 i}$. We multiply the numerator and denominator by $(37-226 i)$ and get the result, but how do I divide it like the normal ...
0
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0answers
25 views

For the ring of natural numbers, what happens if we convert all $k>1$th power of prime numbers to be zero?

So there is a ring of natural numbers. Now someone decides to make all $k>1$th power of prime numbers to be zero. So $2^2 = 2^3 = 2^4 = ... = 3^2 = 3^3 = ... = 0$. Or we can say that these elements ...
1
vote
3answers
82 views

What exactly is a maximal ideal?

I am confused about the definition of maximal ideal. Suppose that there is ring $R$. Now if we select the whole $R$ to be an ideal, then wouldn't this be maximal ideal? Or is the definition of maximal ...
2
votes
1answer
24 views

Nilpotent and Invertible elements in commutative ring with 1

Let $R$ be a commutative ring with $1$, $S$ a subring also with $1$. Suppose $R\setminus S$ contains a nilpotent element. Prove that $R\setminus S$ also contains an invertible element. Attempt at ...
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0answers
25 views

Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
0
votes
1answer
20 views

Image of a Prime ideals in quotient ring

I am having confusion regarding the relationship between the prime ideals of a Ring and a certain quotient of a Ring. Let A be a commutative ring with identity, and a an ideal of A. Clearing there is ...
2
votes
1answer
36 views

Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$.

Let $F$ be a field and let $f(x)$ be a non constant element of $F[x]$. Then, there exists a splitting field $E$ for $f(x)$ over $F$. I have some queries regarding this theorem of existence of ...
1
vote
1answer
16 views

Finitely generated quotient module

In an assignment question, I'm a bit stumped by the following: Let $R$ be a commutative ring, $M$ be an $R$-module and $N$ an $R$-submodule of $M$. I was first asked to prove that if both $N$ and ...
2
votes
3answers
61 views

Contrasting definitions of bimodules? An illusion?

Recently my definition of a bimodule over a $k$-algebra has been challenged and I believe both definitions to be equivalent, am I wrong? Notation: $k$ is a commutative ring and $A$ is a (unital ...
2
votes
2answers
21 views

Torsion elements in integer-modules

In a worryingly short amount of time I've managed to forget almost everything I knew about modules, groups, rings e.t.c. I'm using the definition that an element $m$ of a module $M$ over an integral ...
3
votes
1answer
49 views

How to characterize all finite commutative local rings with the maximal ideal of fixed order (if the order is small)?

Let $R$ be a finite commutative local ring with the maximal ideal $M$ of order $m$. How to characterize all such finite commutative local rings? For examples, if $m=2$, then $R\cong\mathbb{Z}_4$ ...
0
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1answer
40 views

Can a ring be both commutative and anticommutative?

As the title says, can a ring ever be both commutative and anti-commutative? By anti-commutivity, $a \cdot b+b \cdot a =0$. Edit: A ring must have infinitely many numbers.
2
votes
1answer
51 views

Ring structure on $\mathbb{Z}$

Is there a possibilty to define a (nontrivial) ring structure on $\mathbb{Z}$ (or $\mathbb{Q}, \mathbb{R}$) other than the usual so that $\mathbb{Z}$ (or $\mathbb{Q}, \mathbb{R}$) with that structure ...
2
votes
2answers
41 views

Is the ideal $(2,x^4+x^2+1)<\mathbb{Z}[x]$ maximal?, principal?

I'm trying to solve the following problem: Let I=$(2,x^4+x^2+1)<\mathbb{Z}[x]$ be an ideal. Is $I$ maximal? Is $I$ principal? Any help would be appreciated.
2
votes
1answer
142 views

Is normal extension of normal extension always normal?

Let F be a char 0 field, K be a normal extension of F and L be a normal extension of K. Can it be proved or disproved that L is normal extension of F ?
1
vote
1answer
41 views

Showing a matrix is nilpotent if its charateristic polynomial is $t^n$ mod ${\rm nil}(R)$

Let $R$ be a commutative ring. How to prove the following: If $\chi_A(t) \equiv t^n \bmod\operatorname{nil}(R)$ then $A \in M_n(R)$ is nilpotent. Note $\chi_A$ is the characteristic polynomial ...