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
1answer
269 views

Rings' second isomorphism theorem

I am thinking about the proof of the second isomorphism theorem, and something isn't very clear to me. Let $R$ be a ring ,$S\subset R$ a subring and $I\subset R$ an ideal. We have the natural ...
6
votes
3answers
452 views

Can you have a ring homomorphism from a ring to itself which isn't the identity?

By a ring I mean a ring with a multiplicative identity. To me, at this point, this sounds like a fairly simple question, but I haven't been able to come up with any such homomorphism, nor has ...
4
votes
1answer
847 views

Confusion between principal ideal and ideal

Artin defines an ideal $I$ as : $I$ is a subgroup of $R^+$ If $a \in I$ and $r \in R$ , then $ra \in I$ And Principal Ideal is defined as "In any ring, the set of multiples of a particular ...
1
vote
1answer
107 views

Matrix Rings over Artinian commutative Rings

Let $R$ be a commutative artinian ring with identity. It is true that for $n>0$ the matrix ring $M_n(R)$ is left and right artinian?
3
votes
3answers
262 views

Algebraic Elements and Fields of Quotients

The algebraic elements of $\mathbb{R}$ are those elements which are roots of nonzero polynomials with coefficients in $\mathbb{Q}$. In fact, by multiplying through by denominators, we can even take ...
3
votes
1answer
229 views

Characterize finite dimensional algebras without nilpotent elements

Characterize all finite dimensional algebras (may not be commutative) over a field $K$ without nilpotent elements. My condition: Let $A$ be any algebra (may not be finite dimensional), then it's ...
1
vote
1answer
107 views

Commutants of commutative algebras

Let $W$ be a unital algebra and let $V$ be its maximal abelian subalgebra. Must the commutant $V^\prime$ of $V$ be commutative?
3
votes
1answer
287 views

Intersection of principal ideals

An intersection of principal left ideals need not be principal but incidentally this phenomenon is witnessed in von Neumann regular rings. How about arbitrary intersections of infinitely many ...
2
votes
2answers
113 views

homomorphisms and product rings

The problem is this: Let $f:\mathbb{R}[x]\rightarrow \mathbb{C}\times \mathbb{C}$ be the homomorphism defined by $f(x)=(1,i)$ and $f(r)=(r,r)$, for $r\in \mathbb{R}.$ Determine the kernel and the ...
11
votes
3answers
673 views

Are these two quotient rings of $\Bbb Z[x]$ isomorphic?

Are the rings $\mathbb{Z}[x]/(x^2+7)$ and $\mathbb{Z}[x]/(2x^2+7)$ isomorphic? Attempted Solution: My guess is that they are not isomorphic. I am having trouble demonstrating this. Any hints, as to ...
1
vote
2answers
85 views

Small step in proving Hilbert's (Weak) Nullstellensatz

I'm reading up on a proof of Hilbert's Nullstellensatz which uses the Artin-Tate lemma. I followed all of it except for one step, which is probably quite elementary, but my brain may be too fried from ...
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
227 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
203 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
72 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
533 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
193 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 ...