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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4
votes
2answers
395 views

What is true for a ring with exactly two right ideals

This is question #66 from http://www.ets.org/s/gre/pdf/practice_book_math.pdf Let $R$ be a ring with a multiplicative identity. If $U$ is an additive subgroup of $R$ such that $ur \in U$ for all ...
3
votes
3answers
395 views

Analogy for Cosets, Ideals, and Quotient Rings

Edit: FYI my textbook starts with rings instead of groups so I don't really understand Normal Subgroups either. Hi all, I have read the definition of Ideals and Quotient rings many times but don't ...
3
votes
1answer
75 views

Rings and unity

The set $R = {([0]; [2]; [4]; [6]; [8])}$ is a subring of $Z_{10}$. (You do not need to prove this.) Prove that it has a unity and explain why this is surprising. Also, prove that it is a field and ...
1
vote
2answers
97 views

proof that $S=${$w\in R:xw=1$} is an infinite set in a ring $R$ with $xy=1,yx\neq1$ [duplicate]

Let $R$ be a ring and let $x , y$ be elements of $R$ with $xy=1,yx\neq1$. Let $S$ be the set of all elements $w\in R$ such that $xw=1$. How I can proof that $S$ is an infinite set ...
8
votes
1answer
125 views

Does there exist an ordered ring, with $\mathbb{Z}$ as an ordered subring, such that some ring of p-adic integers can be formed as a quotient ring?

I'm not sure if this might instead be MathOverflow material, but I'll give it a shot here first anyway. Does any ring $R$ exist that satisfies the following properties? $R$ is a totally ordered, ...
4
votes
1answer
176 views

Finding Units in a subring of $ \mathbb Q\ $

Consider the set $$R= \left\{\frac{n}{3^k}: n\in\mathbb{Z}\mbox{ and }k\in\mathbb{N}\right\}$$ which is a subring of the rational numbers, $\mathbb{Q}.$ Find the units in this ring $R$. I wanna ...
1
vote
1answer
61 views

Build Onto mapping of a Field to a field with an Integral Domain

The question is as follows, Let D be an integral domain. Let Q be its field of fractions and $\phi$ : D --> Q be the canonical map of D into Q. Prove that, if D is a field, then $\phi$ is ...
2
votes
1answer
541 views

Commutative ring with unity Proof on the set of units?

the question is as follows (TRUE or FALSE.) If R is a commutative ring with unity, then the set of units in R forms a subring. (If true, give a short proof. If false, give a specic counter-example.) ...
1
vote
1answer
1k views

Prove that if $R$ is an Integral Domain then $S$ is an Integral Domain

Let $R$ be a ring and $S$ be a subring of $R$. Prove that if $R$ is an Integral Domain then $S$ is an Integral Domain. I know that an ID is a commutative ring when for $a,b \in R$ if $ab=0$ then ...
1
vote
1answer
219 views

Prove that a non-zero, non-unit element $a \in R$ is irreducible [duplicate]

Prove that a non-zero, non-unit element $a \in R$ is irreducible iff its only divisors are units of $R$ and elements of $R$ which are associates of $a$ I have a proof below but i'm not sure if it is ...
6
votes
1answer
132 views

Embedding of free $R$-algebras

Let $R$ be any nontrivial commutative unital ring and $I$ and $J$ any sets with $|I|>|J|$. Does there exist an embedding of $R$-algebras $R[x_i; i\in I]\longrightarrow R[y_j;j\in J]$? When ...
46
votes
4answers
2k views

Algebra: Best mental images

I'm curious how people think of Algebras (in the universal sense, i.e., monoids, groups, rings, etc.). Cayley diagrams of groups with few generators are useful for thinking about group actions on ...
2
votes
2answers
108 views

$\mathbb{C}[X,Y]/(Y-X^2)\cong\mathbb{C}[X]$

Let $K$ be a field and $f(X)\in K[X]$. Then we have a well-defined surjective homomorphism $$\varphi: K[X,Y]/(Y-f(X))\to K[X]$$ given by $[g(X,Y)]\mapsto g(X,f(X)$. Someone has ...
5
votes
2answers
357 views

Integral domains such that all proper factor rings are finite

Let $\mathbb Z$ be the ring of rational integers. If $a\in\mathbb Z$ is a non-zero element, then the factor ring $\mathbb Z/(a)$ is finite and has order $|a|$. If $\mathbb Z[i]$ is the ring of ...
0
votes
0answers
100 views

End$(A)$ for $A=\mathbb{Z}$, $ A=\mathbb{Q}$ and $A=\mathbb{R}$

By End$(A)$ I mean the set of all ring endomorphisms of $A$. I would appreciate if you could check if my proof is correct. First, for $\mathbb{Z}$ we have $$\phi \colon \ \mathbb{Z} ...
1
vote
2answers
77 views

What are maximal ideals of $K[t]$?

Let $K[t]$ be the algebra of all polynomials in $t$. What are maximal ideals of $K[t]$? I know that $\langle t \rangle = \{tf \mid f \in K[t]\}$ is a maximal ideal. Are there other maximal ideals? ...
3
votes
1answer
113 views

Idealizer of one-sided ideal and endomorphism

Let $A$ be a ring and let $J$ be a right-sided ideal of $A$. We call the set $I_{A}(J)=\lbrace a \in A \mid aJ\subset J\rbrace$ the idealizer of $J$. The ring $E_{A}(J)=I_{A}(J) /J$ is called the ...
3
votes
1answer
286 views

If a ring element is right-invertible, but not left-invertible, then it has infinitely many right-inverses. [duplicate]

Let $A$ be a ring and $a\in A$ an element that has a right-inverse but does not have a left-inverse. Show that $a$ has infinitely many right-inverses.
3
votes
1answer
71 views

Idealizer of one-sided ideal

Let $A$ be a ring and let $J$ be a right-sided ideal of $A$. We call the set $I_{A}(J)=\lbrace a \in A \mid aJ\subset J\rbrace$ the idealizer of $J$. Show that $I_{A}(J)$ is the largest subring of ...
1
vote
1answer
387 views

Let $R$ be a commutative ring and let $I$ and $J$ be ideals of $R$. Show $IJ$ is an ideal of $R$.

Let $R$ be a commutative ring. For ideals $I$, $J \in R$ define $IJ$ to be the set $\{a_1b_1 +\ldots+a_nb_n : n\in\mathbb N$; $a_i\in I$; $b_j \in J\}.$ Prove that $IJ$ is an ideal in $R$.
4
votes
2answers
99 views

Where do I use the fact that $F$ is algebraically closed in this proof?

I have to do the following. Let $F$ be an algebraically closed field. $I\in F[X_1,...,X_n]$ an ideal. Denote by $S(I)$ the subset in $F^n$ consisting of all $n$-tuples $(a_1,...,a_n)\in F^n$ such that ...
1
vote
0answers
32 views

If a finite set of ideals generates a ring, then so does any set of arbitrary powers of those ideals. [duplicate]

In Lang's Algebra, pg 95 (3rd Revised Ed.), he concludes a proof on the Chinese Remainder Theorem with: In the same vein as above, we observe that if $\mathfrak{a_1},\dots,\mathfrak{a_n}$ are ...
3
votes
2answers
83 views

Subgroup of $\mathbb{Z} \oplus \mathbb{Z}$ that is not a subring?

I just had a test, and one of the questions was to show that there is at least one subgroup of $\mathbb{Z} \oplus \mathbb{Z}$ that is not a subring. I couldn't think of one and still can't, so I ...
2
votes
3answers
147 views

Find ideals of ring

I am stuck with a homework problem. Let $R=\mathbb{Z}[\sqrt{ -3}]$. a) Find an ideal $I$ of $R$ such that $(4) \subsetneq I \subsetneq R$. Explain why the inclusions $\subsetneq$ in my example are ...
1
vote
1answer
54 views

Ring $R[x]$ with ideal $I=P[x]$ that $P$ is maximal ideal of $R$

Let $R$ be an commutative and unitary ring that not field and $P$ is maximal ideal of $R$. Now let $I=P[x]$ be an ideal of $R[x]$. Is $I$ is prime? Is $I$ is maximal?
2
votes
1answer
132 views

If $ f _p(x)=1+x+x^2+ \cdots +x^{p-1} $ where $p$ is prime number, $e$ is any positive integer.

Show that $f_p (x^{p^{e-1}})$ is irreducible over $\mathbb{Q}$. I know it is obviously true for $ p=2$. Please suggest me any hints
2
votes
1answer
123 views

Prove that a nontrivial subring of a ring that is a domain has the same identity

Let $R$ be a ring with an identity element $1_R$ which is a domain. Let $S$ be a nontrivial subring of $R$ with identity element $1_S$. Prove that $1_R = 1_S$.
1
vote
1answer
237 views

Characterization of irreducible elements in integral domains.

Let $R$ be an integral domain. Show that a non zero, non unit element $c$ of $R$ is NOT reducible iff its only divisors are units of $R$ and elements of $R$ which are associates of $c$.
1
vote
1answer
108 views

Ring/Field isomorphism before knowing it's a ring/field

If I know $F$ is a ring/field, and I have $G$ (unkown), but I can find a bijective map: $$\Phi:F\to G$$ such that $\forall a,b\in F.\;\Phi(a+b) =\Phi(a)+\Phi(b).\;\Phi(ab) = \Phi(a)\Phi(b)$ Does that ...
0
votes
1answer
65 views

is $\Bbb {Z} /p\Bbb{Z} [x] / (x^2+1) = \Bbb {Z} /p\Bbb{Z}[i]$?? p is prime

let $p$ be prime $\Bbb {Z} /p\Bbb{Z} [x] / (x^2+1)$ any coset will be of the form $ax+b$. and $x^2 + 1 = 0$ so it is important that $x^2 = -1 = p-1$ $(ax+b)(cx+d) =(acx^2 + bd) + x(ad+bc) = ...
3
votes
2answers
205 views

$\Bbb Z_6[x] / (x^2+5)$ quotient ring

I am trying to understand quotient rings and it would be really helpful if someone could show me a general way of mapping quotient rings to simpler rings. Consider $\Bbb Z_6[x] / (x^2+5)$. I know ...
1
vote
0answers
132 views

Showing that a Boolean algebra is a Boolean ring

I've proved that a Boolean ring is a Boolean algebra but I am having trouble with the converse. The operation for + is defined as the symmetric difference for elements $a$ and $b$ from the Boolean ...
4
votes
2answers
231 views

Product of a non-unit with any other element

In any ring, is the product of a non-unit with any other element necessarily a non-unit?
2
votes
1answer
234 views

localization of a module and annihilators

I've just started reading on my own about localizations of modules. I've run into a difficulty as follows: Let $R$ be a commutative ring with unity, $M$ an $R$-module, $S\subseteq R$ a multiplicative ...
1
vote
2answers
94 views

Question about idempotents.

Let $A$ be a $K$-algebra and $B=A/\operatorname{rad} A$, where $\operatorname{rad}A$ is the radical of $A$ (intersection of all maximal right ideals of $A$). Let $e$ be an idempotent of $A$ and ...
1
vote
0answers
47 views

Isomorphisms betweenVerma modules over a semisimple Lie algebra

Fix a finite dimensional, semisimple Lie algebra $L$ and denote the Verma $L$-modules by $V(\lambda ')$ where $\lambda '$ are corresponding weights. Assume that there is an isomorphism between two ...
1
vote
1answer
53 views

Question about modules.

Let $A$ be a $K$-algebra and $B=A/\operatorname{rad} A$, where $\operatorname{rad}A$ is the radical of $A$ (intersection of all maximal right ideals of $A$). Let $I$ be an ideal of $B$ and $S$ be a ...
2
votes
1answer
100 views

Ring homomorphism - isomorphic subring

Please help me with this exercise. Let $K$ be a field, $A$ a ring (without the assumption that a ring must have an identity) and $\phi : K \longrightarrow A$ an homomorphism. Prove either $A$ must ...
3
votes
1answer
263 views

Proof about Noetherian rings

I have to prove that every finite ring is Noetherian. I know examples of Noetherian rings which are not finite such as the field of complex numbers or a PIR like the integers. But anyway: [Proof]: I ...
3
votes
3answers
116 views

If $M$ is an $R$-module and $I\subseteq\mathrm{Ann}(M)$, then $M$ has the structure of an $R/I$-module

Let $M$ be an $R$-module and let $I$ be an ideal of $R$ such that $I$ is a subset of $\mathrm{Ann}(M)$. Define a product of an element of $R/I$ by an element of $M$ as follows: ...
2
votes
1answer
171 views

Reduced Gröbner Basis for $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$

I have $u:=X^2+2XYZ$ and $v:=XY+2Y^2Z-1$ with lex $X>Y>Z $. I have calculated the Gröbner Basis as $G=\{ X^2+2XYZ, XY+2Y^2Z-1, X, 2Y^2Z-1 \}$. But the question I have asks for the Reduced ...
5
votes
2answers
332 views

Equivalent definitions for projective modules

Fact: Let $R$ be a ring with identity. Let $J$ be an $R$-module. Then, $J$ is injective iff for every left ideal $L$ of $R$ every $R$-module homomorphism $L\rightarrow J$ can be extended to an ...
1
vote
2answers
62 views

Integral extensions

Let $p\neq1$ be an integer and let $\beta$ be a root of $x^6-p$. What is the difference, in terms of $\mathbb{Z}$-modules, between $\mathbb{Z}[\beta]$ and $\mathbb{Z}[\beta^2,\beta^3]$? I can ...
1
vote
1answer
26 views

Why $e_1A=M_1$?

Let A be a ring with identity $1$ and $M_1, M_2$ submodules of $A$. We have $1=e_1+e_2$, where $e_i\in M_i$, $i=1, 2$. We can show that $e_i$ are idempotent and $e_1e_2 = e_2e_1 = 0$. We have ...
2
votes
2answers
87 views

$ I(J+L)=IJ+IL$ if $I,J,L$ are ideals of $K$

Given that $I,J,L$ are ideals of $K$, do we have $I(J+L)=IJ+IL$? I am confused how to do it.
1
vote
3answers
359 views

Show that if $R$ is an integal domain, then $R[X]$ is an integral domain.

Let $R$ denote an integral domain, and $R[X]$ denote the polynomials over $R$. Show that $R[X]$ is an integral domain. All I've got left is the non-trivial part - i.e. the cancellation property of ...
1
vote
2answers
160 views

Is $Z(R)$ a maximal ideal?

If $p$ and $q$ are two maximal ideals in the set of zero-divisors in a ring $R$ with non-zero intersection between $p$ and $q$. does the set of all zero-divisors are a maximal ideal and equal the ...
1
vote
6answers
717 views

$ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$ [closed]

How can we prove that $ p^{\frac1n} $ is irrational if $p $ is prime and $n>1$?
0
votes
1answer
46 views

Why $(M/M \operatorname{rad} A) \operatorname{rad}A=0$?

Let $A$ be a ring and $M$ a right $A$-module. Why we have $(M/M \operatorname{rad}A) \operatorname{rad}A=0$? Thank you very much.
2
votes
1answer
34 views

Suppose $R$ is a ring containing a field $F$ in its centre. Construct an injective ring homomorphism from $R$ to $M_n(F)$

Let $R$ be a ring containing $F$ in its center. $R$ is an n-dimensional vector space over $F$, and the homomorphism is to be constructed in terms of a basis of $R$. I'm at a complete loss at what to ...