# Tagged Questions

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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### Examples of commutative rings where the prime subring is not direct summand?

My question consists almost in the title. My motivation is the study of some tensor products $A\otimes_\mathbb{Z} B$. For a (commutative) ring, let us call prime subring the subring generated by $1$ i....
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### A universal construction of the field of fractions of an integral domain?

Let $R$ be an integral domain and For a field $\hat R$ consider the following : There is an injective ring homomorphism $i:R \to \hat R$ such that for any field $F$ and any injective ring homomorphism ...
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### Show that there exist a noncommutative ring (with identity) of order $p^3$.

Let p be a fixed prime. Show that there exist a noncommutative ring (with identity) of order $p^3$. RemarkI was able to $p = 2$: $U_n(\mathbb{Z}_2)$ - the set of $n \times n$ matrices with entries ...
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### Unfamiliar w/ Ring Notation

I'm used to seeing rings represented as sets, but in one of my homework problems, I am asked to: Find the number of zero-divisors of $R_{x^2-x}$. Can somebody please explain what this notation ...
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### Products and relationships of ideals of Ring R.

Let $R$ be a ring and let $I$ be a left ideal of $R$. (a) Let $K$ be a left ideal of $R$. Show that $(IK)^{n} \subseteq I^{n}K$ for all $n \in \mathbb{N}$ (b) Show that $I+ IR$ is a two-sided ideal ...
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### Do partially ordered rings that aren't totally-ordered, which nonetheless satisfy $x+y \geq 0 \rightarrow x \geq 0 \vee y \geq 0,$ exist?

(All my rings are commutative and unital. I include $1 \geq 0$ as a partially-ordered ring axiom.) Let $R$ denote a partially-ordered ring. Observe that if $R$ is totally-ordered, then $R$ satisfies ...
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### Question related to ring theory

Let $p$ be an odd prime and let $1 + \frac{1}{2} + \cdots + \frac 1{p-1} = \frac ab$, where $a,b$ are integers. Show that $p\mid a$. (Hint: As $a$ runs through $U_p$, so does $a^{-1}$.) P.S. $U_p$ ...
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### If f is surjective then f is not a right divisor of zero

Let $R$ be a ring and $M$ a R-module. For $r\in R$ define $f:M\to M$ by $f(s)=sr$. Show that $f$ is injective if and only if $r$ is not a right zero divisor. I have done a similar problem to this in ...
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### What is a linear combination, exactly?

I'm used to the definition of linear combination used in linear algebra textbooks. I'm reading the book Algebra by Artin and on page 357 he says: If $R$ is the ring $\mathbb{Z}[x]$ of integer ...
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### How can I show that any one to one endomorphism of an Artinian module is an automorphism?

How can I show that any one to one endomorphism of an Artinian module $M$ is an automorphism? I was given this question and I presume that it is really to show that Artinian modules are co-hopfian. ...
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### To find all integers $n > 1$ for which $(n-1)!$ is a zero-divisor in $Z_n$.

To find all integers $n > 1$ for which $(n-1)!$ is a zero-divisor in $Z_n$. (Gallian Problem) $Z_n$ does not contain any zero divisors when $n$ is a prime number. So we look at the composite ...
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### Why Can't we Factor Invertible Elements?

I'm currently studying Herstein's Algebra; specifically, UFDs and the abstract notion of factorization. This is perhaps more of an intuitive question than one with a hard answer. We define ...
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### What does $\Bbb{Z}/(2) \times \Bbb{Z}/(3) \times \dots$ do?

Take the product of rings $M = \Bbb{Z}/(2) \times \Bbb{Z}/(3) \times \dots$ over the primes or in general take any infinite set of quotient modules of a ring $R$ and form their product. It's true ...
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### Ring of Infinitely Differentiable Functions

Denote the ring $\text{C}^{\infty}$ i.e. infinitely differentiable functions from $\mathbb{R}\mapsto\mathbb{R}$. I have managed to prove that this ring is a Principal Ideal Domain. I must also prove ...
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### Polynomial ring with integral coefficients is integral

Let $B$ be a ring and $A\subset B$ a subring. Assume that $B$ is integral over $A$. I have to prove that $B[X]$ is integral over $A[X]$. I tried writing down an integral relation for $f(X)\in B[X]$ ...
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### Showing that an element is prime in $\mathbb{Z}$[i]

Let p be a prime integer, and suppose p = a2 + b2 has NO integer solution. The exercise asks that if p = a2 + b2 has no solution, then p is a prime in the set of Gaussian integers $\mathbb{Z}$[i], ...
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### Invertible element vs zero divisor in a ring

Let $R$ be a ring and $x,y\in R$ such that $yx=1$ and $xy\ne 1$. Prove there is $z\ne 0$ such that $yz=zx=0$. My first thought is that y is not an invertible element. Does that mean that it is a ...
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### Show that $End_{\mathbb{K}}(\mathbb{V})$ is Dedekind finite ring.

A ring $R$ is said to be Dedekind finite if $ab=1 \Rightarrow ba=1$. Let $\mathbb{V}$ a finite-dimensional $\mathbb{K}$-vector space, show that $End_{\mathbb{K}}(\mathbb{V})$ is Dedekind finite ring.
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### Given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute.

Let $R$ be a ring with identity. An element $a \in R$ it is idempotent if $a^2=a$. Show that given two idempotents $a,b \in R$ such that $a+b$ is idempotent then $a$ and $b$ commute. Remark: I'...
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### Construction of homomorphism between $^\ast\mathbb{R}$ and $^*\mathbb{Q\cap L}$

Denote by $\mathbb{I}$ the ring of infinitesimals and by $\mathbb{L}$ the ring of finite hyper-reals. Prove that $$\mathbb{R}\cong{^\ast\mathbb{Q\cap L/^\ast Q\cap I}}.$$ I thought using the first ...
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### A question concerning to show that $V(I)$ is open if $I$ is radical ideal

Let $I:=(f_1 ,...,f_k)$ be a finitely generated ideal of $C(X,\mathbb R)$ such that $\mathrm{rad}(I)=I$, $f:=\sqrt{\sum_{m=1}^k |f_m|}=\sum _{m=1}^k g_mf_m$ where $g_i$'s are real valued continuous ...
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### To show that an integer $m$ is prime element in $\Bbb Z[i]$ if $m$ is a prime number of the form $4n+3$.

Let $p$ be a prime number of the form $4n+1$. Then show that $p = a^2 + b^2$ for some $a,b \in \Bbb Z$ and $p$ is not prime in $\Bbb Z[i]$. Also show that an integer $m$ is prime element in $\Bbb Z[i]$...
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### Tensor product of algebras and generating sets.

Let $A$ be a module over $k$ generated by $x$ and $y$. The generating set for $A \otimes_k A$ is $\{x \otimes x, x \otimes y, y \otimes x, y \otimes y\}$. But does this still hold if $A$ is an algebra ...
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### Proving that a surjective homomorphism can help generate a finitely generated k-algebra

I am trying to understand a proof in Dummit and Foote. It can be found in Chapter 15, Section 1, Corollary 5. The corollary is The ring $R$ is a finitely generated $k$-algebra iff there is some ...
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### $I^{j}/I^{j+1} \cong R/I$ for any ideal I in ring R.

Let $R$ be a commutative ring with $1$ and $I$ be an ideal in it. Let $\overline{\alpha} \in I^j\setminus I^{j+1}$ and define $\theta\colon R \to I^j/I^{j+1}$ by $\theta(x)=\overline{\alpha x}$. My ...
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### Show that if $a \in R$ has more than one left-inverse then it has infinite. [duplicate]

Let R is a ring with identity $1$. Show that if $a \in R$ has more than one left-inverse then it has infinite.
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### Ring homomorphisms on the set of rationals that coincide on integers

Let $R$ be a ring and let $f, g: \mathbb Q \to R$ be two ring homomorphisms such that $f|_{\mathbb Z}=g|_{\mathbb Z}.$ Then $f=g.$ I was trying to prove the above mentioned statement. According to ...
Question: Let $R$ be a ring and let $J$ be a left ideal of $R$. Assume that $J$ is nilpotent. Prove that $J$ is contained in a nilpotent 2-sided ideal of $R$. Comments: I have found lots of ...