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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9
votes
2answers
434 views

A non-nilpotent formal power series with nilpotent coefficients

Does anyone have an example of a formal power series $$p=a_0+a_1x+ a_2x^2 + \cdots \in R[[x]]$$ ($R$ is a commutative ring) all of whose coefficients $a_i$ are nilpotent in $R$ such that $p$ is not ...
6
votes
2answers
180 views

Irreducible polynomial over field of order p

Let $p$ be a prime and $F=\mathbb{Z}/p\mathbb{Z}$ and $f(t)\in F[t]$ be an irreducible polynomial of degree $d$. I need to show that $f(t)$ divides $t^{(p^{n})}-t$ if and only if $d$ divides $n$.
3
votes
3answers
217 views

How to find ideals?

Let $R=\mathcal{C}([0,1],\mathbb{R})$ be the ring (standard one) of continuous functions. For each $\gamma\in[0,1]$, let $I_\gamma=\{f\in R; f(\gamma)=0\}$. It is easy to prove that $I_\gamma$ is an ...
4
votes
2answers
197 views

How to determine if this is a field?

A finite subring $R$ of a field $V$ contains $1$ (so $1$ is an element of $R$). The question is: True or False: The ring $R$ must be a field. I thought that if $R$ was a field it had to be a finite ...
2
votes
2answers
220 views

Noncommutative rings, finding $a$ and $b$ such that every term in the sum $(a+b)^n = a^n + a^{n-1}b + \ldots$ is distinct

This question was inspired by the binomial theorem for rings. For commutative rings, we have the identity $$(a+b)^n = \sum_{k=0}^n {n \choose k}a^kb^{n-k}$$ which does not hold for non-commutative ...
2
votes
1answer
64 views

For which $m \in \mathbb N$ is the ideal $(m,x^2+y^2)$ prime in $\mathbb Z[x,y]$?

Let $m \in \mathbb N$. Find a necessary and sufficient condition for $m$ such that the ideal $(m,x^2+y^2)$ is prime in $\mathbb Z[x,y]$. I have to find for which $m$ the quotient ring is an ...
4
votes
2answers
143 views

Localization and Extension of modules

Let $R$ be a commutative ring and $S$ be an $R$-algebra. Assume that $S$ is finitely generated as an $R$-module. Let $M$ and $N$ be finitely generated $S$-modules and $\mathfrak{m}$ a maximal ideal ...
9
votes
1answer
270 views

Tensor product of simple modules

Let $M$ a right simple module and $N$ be a left simple module over a ring $R$. My questions are: How can we describe $M \otimes_R N$ explicitly? Well, I guess that it is a quotient of $R$ by a sum of ...
1
vote
1answer
66 views

Whether there is a non commutative k-algebra of dimension no larger than 3

Suppose $k$ is a field and $A$ is a $k$-algebra of dimension no larger than 3. If $A$ is semi-simple, then $A$ can be written as a direct sum of simple $k$-algebras. Further one can find $A$ is ...
2
votes
1answer
59 views

How many products to be a ring?

I got the question below studying this problem: $p$-Sylow subring. Let $(R_1,+_1,\cdot_1)$ and $(R_2,+_2,\cdot_2)$ be two rings with identity elements $e_1,e_2$. Let $(R,+)$ be the group defined by ...
2
votes
1answer
151 views

Name a ring of 2 by 2 matrices where $a^3 = a$ and a belonging to this ring?

I need an example of a ring consisting of 2 by 2 matrices where $a^3=a$ with $a$ belonging to this ring. If someone can list the elements I would be satisfied. What I'm trying to get at it is ...
1
vote
2answers
202 views

How can we compute the power of an ideal?

Let $I$ be an ideal in a Noetherian ring $R$ which is generated by $x_1,...,x_n$. From this system, can we find out what is the generating set for an arbitrary power of $I$: $I^k$? Is it ...
4
votes
1answer
71 views

$p$-Sylow subring

I would like to know if there is some results concerning about the following question: When could a $p$-Sylow subgroup of a finite ring $R$ be a subring? In other words, is it possible to induce ...
-2
votes
1answer
114 views

Annihilator of a simple module 2 [duplicate]

Possible Duplicate: Annihilator of a simple module Let me ask the same question as before because I still have trouble understanding the problem. Let $R$ be a finitely generated ...
1
vote
3answers
180 views

Are monomorphisms of rings injective?

Let $R$ and $S$ be rings and $f:R\to S$ a monomorphism. Is $f$ injective?
2
votes
4answers
103 views

Need help to show $R/I$ is not necessarily flat over $R$

Let $R$ be a ring with unit and $I$ an ideal in $R$. I want to show that $R/I$ is need not be flat over $R$, but I do not know how to come up with a counter-example. Any hint is appreciated.
1
vote
2answers
74 views

About absolute convergence and completeness in rings

EDIT: Let R be a commutative ring with unit ring and $I$ a maximal ideal in R. The completion of R with respect to $I$ is the inverse limit of the factor rings $R / I^k$ under the usual quotient maps. ...
3
votes
1answer
163 views

Commutative Algebra - Polynomial Rings

Let $Z$ be the ring of integers, $p$ a prime and $F_p = Z/pZ$ the field with $p$ elements. Let $x$ be an indeterminate. Set $R_1 = F_p[x]/(x^2-2)$, $R_2 = F_p[x]/(x^2-3)$. Determine whether the rings ...
5
votes
3answers
188 views

How to construct a $2\times 2$ real matrix $A$ not equal to Identity such that $A^3=I$?

How to construct a $2\times 2$ real matrix A not equal to Identity such that $A^3$=I? There is a correspondence between the ring of complex numbers and the ring of $2\times2$ matrices (0 matrix is ...
2
votes
1answer
532 views

On modules over polynomial rings

Let $\mathbb{A}$ be a polynomial ring in $n$ variables over an algebraically closed field $\mathbb F$. Given a maximal ideal $\mathfrak{m}$ of $\mathbb A$, consider the quotient ...
2
votes
1answer
209 views

Relaxing the definition of a von Neumann regular ring

Hereinafter, all rings are assumed to be unital but not necessarily commutative. A well-known class of rings are von Neumann regular rings, that is, rings $R$ such that for each $a\in R$ there is an ...
4
votes
2answers
122 views

Finite presentation of algebra of invariants

(1) Let $R$ be a ring, let $A$ be a finitely presented $R$-algebra, and let $G$ be a finite group of $R$-automorphisms of $A$. Is the algebra of invariant $A^G$ finitely presented over $R$? I can ...
2
votes
1answer
127 views

If $R$ is a ring s.t. $(R,+)$ is finitely generated and $P$ is a maximal ideal then $R/P$ is a finite field

Let $R$ be a commutative unitary ring and suppose that the abelian group $(R,+)$ is finitely generated. Let's also $P$ be a maximal ideal of $R$. Then $R/P$ is a finite field. Well, the ...
6
votes
2answers
484 views

Every Ring is Isomorphic to a Subring of an Endomorphism Ring of an Abelian Group

Show that for every ring $(R,+,\cdot)$, there is an abelian group, $(A,+)$, such that $R$ is isomorphic to a subring of $(\operatorname{End}(A),+,\circ)$. $(\operatorname{End}(A),+,\circ)$ is the set ...
8
votes
1answer
99 views

Are there any examples of rings $R$ such that $\mathrm{End}(R,+,0)\not\cong R$?

In a handful of examples, I've noticed that the endomorphism ring $\mathrm{End}(R,+,0)$ is isomorphic to the ring $R$ itself. For instance, $\mathrm{End}(\mathbb{Z},+,0)\cong\mathbb{Z}$ and ...
37
votes
4answers
2k views

What kind of work do modern day algebraists do?

Often times in my studies I get the impression that algebra is just a tool to help with other branches of mathematics, like algebraic geometry, algebraic number theory, algebraic topology, etc. How ...
2
votes
0answers
84 views

Projective dimension of simple module

Let $R$ be a commutative ring and $M$ a simple $R$-module. Then $\mathfrak{m}=Ann(M)$ is a maximal ideal of $R$. Then it is known that $$ \mathrm{pdim}_{R}(M)=\mathrm{pdim}_{R_{\mathfrak{m}}}(M), $$ ...
4
votes
1answer
123 views

Automorphism groups of real clifford algebras

I'm sure someone has already worked-out what all the relevant groups really are; my question is about how signature duality interacts with these groups. So, by an awful calculation, and choosing a ...
2
votes
3answers
292 views

Are all units of a ring associates?

When studying UFDs I started to get confused... If $u$,$v$ are units in $R$ then $u^{-1}$$v$ is a unit in $R$ and so $v$ = ($u$$u^{-1}$)$v$ = $u$($u^{-1}$$v$) hence u and v are associates..? Are ...
2
votes
2answers
171 views

Spectrum of an element in sub-algebra: $\sigma_A(b)\setminus \{0\}\subseteq \sigma_B(b) \setminus \{0\}$

Please help me to prove this:(or give me some references for this.) Thanks very much! Let $A$ be a (unital) algebra and $B\subset A$ a (unital) sub-algebra. Then for all $b\in B$: ...
5
votes
3answers
252 views

Modules over Local rings

Let $R$ be a finite commutative local ring with identity. If $M$ is a finite $R$-module it is necessarily projective?
1
vote
4answers
183 views

Global dimension of free algebra.

Is there any easy way to see the global dimension of a free algebra $$ A=k\langle x_{1},\dots,x_{n} \rangle $$ is 1?
5
votes
2answers
347 views

Ring homomorphism with $\phi(1_R) \neq1_S$

Let $R$ and $S$ be rings with unity $1_R$ and $1_S$ respectively. Let $\phi\colon R\to S$ be a ring homomorphism. Give an example of a non-zero $\phi$ such that $\phi(1_R)\neq 1_S$ In trying to find ...
1
vote
2answers
50 views

$im(I)=im(R)$ implies what?

I'm studying an ideal $I \trianglelefteq R$ and noticed that for a certain non-injective, non-zero homomorphism $\varphi: R \rightarrow S$ I can show that $\varphi(I)=\varphi(R)$. I'm wondering if ...
0
votes
1answer
248 views

Nilpotent Element And Jacobson Radical

I am looking ring with nilpotent element such that $J(R)=0$ where $J(R)$ is Jacobson radical. Any suggestion?
2
votes
1answer
228 views

Residue field of polynomial rings

Let $k$ be an algebraically closed field of characteristic $p$ and $A=k[x_1,\cdots,x_n]$ the polynomial ring over $k$ in $n$ variables. Given a prime ideal $\mathfrak{p}$ in $A$, denote by ...
4
votes
1answer
192 views

What are some examples of vector spaces that aren't graded?

From wikipedia: a vector space $V$ is graded if it decomposes into direct sum $ \oplus_{n \geq 0} V_n$ of vector spaces $V_n$. So as far as I understand things, any vector space with a countable ...
0
votes
1answer
139 views

Integral domain problem

Let $D$ be an integral domain. Prove that every automorphism of $D[x]$ is of the form: $\phi_{a,b} : D[x] \rightarrow D[x]$ $f$ $\rightarrow$ $f(ax+b)$ where a is a unit of $D$ and $b \in D$. ...
1
vote
0answers
151 views

All finite-dimensional simple modules are $1$-dimensional

Let $A$ be a (non-commutative) $k$-algebra, where $k$ is an algebraically closed, characteristic zero field. Let $M$ be a finite-dimensional simple $A$-module. If $A/\operatorname{ann}(M)$ is ...
5
votes
1answer
262 views

Invertible elements in the ring $K[x,y,z,t]/(xy+zt-1)$

I would like to know how big is the set of invertible elements in the ring $$R=K[x,y,z,t]/(xy+zt-1),$$ where $K$ is any field. In particular whether any invertible element is a (edit: scalar) multiple ...
0
votes
1answer
84 views

Graded Ring - Finite Sum

I've just read that if $R=R_0\oplus R_1 \oplus \dots$ is a graded ring and $f\in R$ then there's a unique decomposition of $f$ as $f=f_0+\dots+f_n$ with $f_i\in R_i$. I can't see immediately why in ...
3
votes
1answer
178 views

Localization of a non-commutative ring

Let $A$ be the non-commutative ring given by $$ A=\mathbb{C}\langle x,y,z \rangle /(xy=ayx,yz=bzy,zx=cxz) $$ for some $a,b,c\in \mathbb{C}$. What is the localization $A_{(x)}$ of A with respect to the ...
0
votes
2answers
102 views

Question about isomorphism between a ideal and a polynomial ring

Sorry for my ignorance, my question is: Let be $F[X]$ a polynomial quotient ring, where $F$ is a finite field with characteristic 2. Are there any ideal, $I$, such that $I$ is isomorphic to $F[X]$?.
3
votes
1answer
140 views

Why is $S/Z$ a domain for the ideal $Z=\{z\in S\mid za=0,\;\forall a\in R\}$ in $S$?

Suppose $R$ is a rng with no zero divisors, not necessarily commutative. I know $R$ can be embedded into a ring $S:=\mathbb{Z}\times R$ by identifying $r\in R$ with $(0,r)\in S$. The operations on $S$ ...
2
votes
1answer
64 views

Equality in rng with no zero divisors.

I'm working on this problem, but I'm missing some manipulation. Suppose $R$ is a rng without zero divisors and has elements $a$ and $b\neq 0$ such that $ab+kb=0$ for some $k\in\mathbb{N}$ (that is, ...
2
votes
1answer
94 views

A question on artinian semi-primitive rings

So the question is as follows: Suppose $U$ is an ideal of artinian ring $R$, then show that there is an ideal $V$ such that $U+V=R$ and $U\cap V \subseteq J(R)$ . Let me describe my approach. I took ...
3
votes
1answer
176 views

Theorem of Kaplansky, $R$ is a division ring if every element but one is (right) quasi-invertible.

There is a theorem of Kaplansky that seems to pop up every algebra book. Here rng denotes a ring with possibly no identity. As definition, an element $a$ of a rng $R$ is said to be (right) ...
6
votes
1answer
171 views

Are minimal prime ideals in a graded ring graded?

Let $A=\oplus A_i$ be a graded ring. Let $\mathfrak p$ be a minimal prime in $A$. Is $\mathfrak p$ a graded ideal? Intuitively, this means the irreducible components of a projective variety are ...
2
votes
1answer
122 views

A problem on the Jacobson radical, from Isaacs Graduate Algebra

This is problem 14.10 from Isaacs Graduate Algebra. Let $U$ and $V$ be ideals of a ring $R$ and assume $U+V$ = $R$, and $U \cap V \subseteq J(R)$ . Suppose that $v \in V$ and that $U + v$ is ...
3
votes
2answers
119 views

$B \otimes_A A[X] \cong B[X]$

Let $A$ be a subring of a commutative unital ring $B$. Can you tell me if my proof of the following claim is correct? Claim: $B \otimes_A A[X] \cong B[X]$ Proof: It's enough to show that $B[X]$ ...