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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3
votes
2answers
56 views

Show there is exactly one ring homomorphism $g: R[x] \to S[x]$ s.t. $g(r)=f(r)$

give a ring homomorphism $f:R \to S$ and $p \in S[x]$ Show there is exactly one ring homomorphism $g: R[x] \to S[x]$ s.t. $g(r)=f(r), \forall r \in R$ and $g(x)=p$ So whatever $f$ does I'm not sure ...
0
votes
1answer
61 views

need help with a proof, Eulers theorem

I will not post the entire proof here. Just one part of the proof that they seem to use, then it will be simpler for you to read. They use that All the elements of $\mathbb{Z}_n$ that are relatively ...
0
votes
1answer
83 views

Projective dimension of module over local ring

This question arose reading the well known article by Buchsbaum Lectures on regular local rings. He states without proof that, given $(R,m)$ a local ring and an $R$-module $M$ over $R$, we have ...
0
votes
1answer
79 views

A case where a UFD is a PID

Let $R$ be a unique factorization domain (UFD). Prove that if $R$ is such that every ideal generated by two elements is principal, then $R$ is a principal ideal domain (PID). I'm having some trouble ...
8
votes
1answer
906 views

Upper Triangular Block Matrix Determinant by induction

We want to prove that: $$\det\begin{pmatrix}A & C \\ 0 & B\\ \end{pmatrix}= \det(A)\operatorname{det}(B),$$ where $A \in M_{m\times m}(R)$, $C \in M_{m\times n}(R)$,$B \in M_{n\times n}(R)$ ...
0
votes
0answers
27 views

$\sim$ on $R \times (R \backslash \{0\})$ show it is an equivalence relation and a congruence [R is an integral domain]

$\sim$on $R \times (R \backslash \{0\})$ by $(a,b) \sim (c,d)$ means $ad=bc$ Note:$R$ is an integral domain The operations are $(a,b)+(c,d) = (ad+bc,bd)$ and $(a,b) \cdot (c,d) = (ac, db)$ ...
1
vote
1answer
76 views

Does canonical projection of commutative rings(R to R/I) always send prime ideals to prime ideals?

R is a commutative ring with 1, I is an ideal of R. Consider the canonical projection f: R to R/I. Suppose p is a prime ideal of R then is f(p) always prime? I think if ab+I$\in$f(P) with ab$\in$ p, ...
2
votes
1answer
63 views

Extending scalars from $k$ to $K$: how find $K$ linear maps?

If I extend scalars from a ring $R$ to a ring $S$ by a homomorphism $f:R \to S$, then starting with an $R$ module $M$, I get an $S$ module $S \otimes_R M$. Given $\sigma \in \text{End}_R(M)$, I know ...
13
votes
4answers
1k views

An example of a Ring with many zero divisors

Is there an example of a commutative ring $R$ with identity such that all its elements distinct from $1$ are zero-divisors? I know that in a finite ring all the elements are units or zero-divisors. ...
0
votes
1answer
118 views

Do non-graded rings exist?

I've been reading about graded rings recently, and I was wondering if there exist commutative or non-commutative rings for which no non-trivial gradings exist? That is, a ring for which if $R=\...
3
votes
1answer
48 views

One-sided nilpotent ideal not in the Jacobson radical?

Problem XVII.5a of Lang's Algebra, revised 3rd edition, is: Suppose $N$ is a two-sided nilpotent ideal of a ring $R$. Show that $N$ is contained in the Jacobson radical $J: = \{ \cap\, I: I \...
3
votes
1answer
93 views

Intersection of ring and prime ideal

Give an example of an extension $B/A$ of rings, with $B$ an integral domain and a nonzero prime ideal $\mathfrak{p}$ of B such that $\mathfrak{p} \cap A=(0).$ I don't know where to begin with this.. ...
1
vote
1answer
167 views

Order of group of units of ${\bf Z}_{2015}[X]$

I know that the order of the group of units $U({\bf Z}_{2015})$ is 1440 and so the order of the group of units of the polynomial ring ${\bf Z}_{2015}[X]$ must be at least that because we can view each ...
-3
votes
1answer
162 views

By applying the Fundamental Theorem of Homomorphisms, show that there is a ring isomorphism $g: \mathbb{Z}_{4} \to \operatorname{im}(f)$

Please refer to the question here for additional details. Theorem: If $R$ and $S$ are rings and $\phi: R \to S$ a ring homomorphism defined by $g(n+\ker(\phi)) = f(n)$, then $R/Ker(\phi) \cong \...
0
votes
1answer
82 views

Principal ideal domain with finitely many ideals

Let $aR$ be a nonzero ideal in a PID $R$. Show that $R/aR$ is a ring with only finitely many ideals. Honestly, I do not know how to start. Appreciate any tips.
3
votes
0answers
153 views

Find the $\ker(f)$ and $\text{Im}(f)$.

Consider the rings $\mathbb{Z}$, $\mathbb{Z}_{4} = \{\bar{0},\bar{1},\bar{2},\bar{3}\}$ and $\mathbb{Z}_{12} = \{[0],[1],[2],...[11]\}$. Define $f: \mathbb{Z} \to \mathbb{Z}_{12}$ by $f(x) = 9x$. ...
0
votes
2answers
77 views

What do Ideals tell you? [duplicate]

So I'm revising definitions of algebra for my exam and I'm wondering what an Ideal actually is? I believe the definition is: $I$ is an ideal of $R$ if $xr,rx\in I$ where $r\in R$ and $x\in I$ ...
0
votes
2answers
747 views

new addition and new multiplication x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z, prove the set Z equipped with these 2 new operation

here says a new operation addition and new operation multiplication x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z,where the operations on the right hand side are ordinary addition and multiplication ...
0
votes
1answer
52 views

Quotient Rings of Algebras

So let us take the commutative Banach algebra $B=\mathscr{l}^1(\mathbb{Z}_n)$ over $\mathbb{R}$ with convolution as multiplication $(f\ast g)(x)=\underset{y\in \mathbb{Z}_n}{\sum} f(y)g(x-y)$. I know ...
2
votes
0answers
29 views

Example of a maximal idea

Let $A$ be the set of bounded continuous functions from the set of real numbers to itself. Then $A$ is a ring under pointwise addition and multiplication. The set $I$ of all functions $f \in A$ ...
0
votes
2answers
132 views

$\mathbb{Z}[x_{1},\dots,x_{n}]/I$ is a field therefore it's finite [duplicate]

I'd spent much time for this but didn't get any results.. Could u give me only the idea but not a full proof
2
votes
1answer
48 views

Prove $I \subseteq I+J$ and $J \subseteq I+J$

Let $R$ be a ring and $I$ and $J$ be the ideals of $R$. Prove Prove $I \subseteq I+J$ and $J \subseteq I+J$ I know this is very trivial, but I still need to check what I am doing is correct or not... ...
0
votes
1answer
37 views

Finitely generated as an Algebra

Let $R,S$ be rings. Is the following equivalent to saying $S$ is finitely generated as an $R$-algebra? "For some $n \in \mathbb{N} $ there exists a surjective ring homomorphism from $R[t_1,\dots,t_n]$...
0
votes
1answer
109 views

Why doesn't there exist a ring homomorphism between these two rings?

Why doesn't there exist a ring homomorphism between $\mathbf{Q}[x]/(x^2-2) $ and $\mathbf{Q}[x]/(x^2 -3) $? I see both rings are in fact fields as the polynomials are irreducible, further I know for $...
0
votes
2answers
134 views

Is $\phi: a + b \sqrt{2} \rightarrow a + b\sqrt{3}$ a $\mathbb Q(\sqrt{2}) \rightarrow \mathbb Q(\sqrt{3})$ field isomorphism?

Is $\phi: a + b \sqrt{2} \rightarrow a + b\sqrt{3}$ a $\mathbb Q(\sqrt{2}) \rightarrow \mathbb Q(\sqrt{3})$ field isomorphism? Some book I'm reading says so. But I'm a bit lost: for field ...
0
votes
3answers
73 views

Prove/disprove: $I \cup J$ is (always) an Ideal of $R$.

Let $I$ and $J$ be the ideals of $R$. Prove/disprove: $I \cup J$ is (always) an Ideal of $R$. Rough Sketch: Since, $I$ and $J$ are the ideals of $R$, we have $0_R \in I$ or $0_R \in J$. Hence, $0_R \...
0
votes
1answer
30 views

Manipulations of Euclidean domains

I am trying to answer the following question For (a) I have said that a and ab are in the ring R, by the definition of a ring. Therefore, by the definition of a Euclidean domain a=abq+r. As we are ...
0
votes
2answers
104 views

The quotient of product of rings

Assume $R_1$ and $R_2$ are two commutative rings with identities, their direct product ring is $R_1\times R_2$, and its ideal can be in the form of $I_1\times I_2$. When considering $R_1\times R_2$/$...
1
vote
1answer
40 views

Show that the ring $A(U)=A_f$ depends only on $U$ and not on $f$.

Let $A$ be a ring and let $X=Spec(A)$ and let $U$ be a basic open set in $X$. (i.e. $U=X_f$ for some $f∈A$). If $U=X_f$, show that the ring $A(U)=A_f$ depends only on $U$ and not on $f$. My Work: ...
4
votes
1answer
55 views

Ring of continuous functions is integral over a subring

Is the ring of all continuous functions $\mathbb{R}^2 \to \mathbb{R}$ integral over the subring of functions $f$ such that $f(1,0) = f(0,1)$?
2
votes
1answer
182 views

To find the nilpotent elements of $\Bbb Z_n$ and also the number of nilpotent elements of $\Bbb Z_n$.

I am trying to find the nilpotent elements of $\Bbb Z_n$ and also the number of nilpotent elements of $\Bbb Z_n$. My Try: If $\bar a$ be a nilpotent element then there exists a $k \in \Bbb Z$ such ...
1
vote
2answers
66 views

To prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups. [duplicate]

Let $(F,+,\cdot)$ be a field. Then to prove that $(F,+)$ and $(F-\{0\},\cdot)$ are not isomorphic as groups. I am facing difficulty in finding the map to bring a contradiction!!
2
votes
2answers
61 views

To show that $7 \Bbb Z$ and $16 \Bbb Z$ are isomorphic as groups but not isomorphic as rings.

To show that $7 \Bbb Z$ and $16 \Bbb Z$ are isomorphic as groups but not isomorphic as rings. I have done the first part but finding difficult to show that they are not isomorphic as rings??
3
votes
0answers
35 views

Efficiently computing GCDs in $\mathbb{Z}[(1+\sqrt{-19})/2]$

The ring $\mathbb{Z}[(1+\sqrt{-19})/2]$ is a PID; hence any two elements have a GCD. How you would compute their GCD? In a Euclidean domain, you would use the Euclidean algorithm. But $\mathbb{Z}[(1+\...
2
votes
3answers
226 views

how to prove $R[x]/I [x]=(R/I)[x]$?

If $I$ is an ideal of a ring $R$, then prove that $R\left[x\right] / \left(I\right) \cong \left(R/I\right)\left[x\right]$ (where $x$ is a polynomial indeterminate). Here, $\left(I\right)$ denotes the ...
1
vote
1answer
55 views

Characterization of prime ideals of $S^{-1}R$ when $S=1+I$, $I$ an ideal?

How can we characterize the prime ideals of $S^{-1}R$ when $S=1+I$, and $I$ is an ideal? Clearly if $p$ is a prime containing $I$ then $S^{-1}p$ is a prime of $S^{-1}R$
3
votes
1answer
66 views

Derived categories of filtered modules

For a ${\mathbb Z}$-filtered ring ${\mathbb k}$ one can consider the category ${\mathbb k}\text{-filt}$ of ${\mathbb Z}$-filtered ${\mathbb k}$-modules, equipped with the exact structure which ...
1
vote
1answer
19 views

For every $n \in \mathbb Z^+$ , does there exist a ring automorphism $\psi$ on $\mathbb C[x]$ of order $n$?

For every $n \in \mathbb Z^+$ , does there exist a ring automorphism $\psi$ on $\mathbb C[x]$ such that $\psi^n=\psi \circ... n $ times is the identity automorphism but $\psi^r$ is not identity for ...
3
votes
1answer
138 views

How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$?

How to find all ring homomorphisms from $\mathbb Z_{12} \to \mathbb Z_{30}$? I know that it is enough to determine $f([1]_{12})$ ; moreover $f([1]_{12})$ should be an idempotent element of $\mathbb ...
-1
votes
1answer
78 views

Ring theory, isomorphism proof. [closed]

Let $R=\{(a,b)∈Z\times Z: a\equiv b \mod 2\}$. I need to prove that $Z[x]/(x^2-1)$ is isomorphic to $R$. Can anybody give me a hint?
0
votes
3answers
64 views

Q[x,y]/(x) = Q[y]? [closed]

I tried to use R[x]/I =(R/I)[x] to prove it and somewhat done. The thing is tho i'd like to see how it should be written formally and how to interpret terms such as (Q[y])[x]/ (x) provided it is ...
2
votes
1answer
74 views

Factorization of a prime ideal in a integral extension.

If $R\subseteq R'$ are integral extensions of Dedekind rings, and $0\neq\mathfrak p$ is a prime ideal of $R$ then $R'\mathfrak p\neq R'$. Do you know an example $R'\mathfrak p=R'$?. Of course $R\...
2
votes
3answers
80 views

How to go about proving that $-1+(x-4)(x-3)(x-2)(x-1)$ is irreducible in $\mathbb{Q}$?

How do you show that $-1+(x-4)(x-3)(x-2)(x-1)$ is irreducible in $\mathbb{Q}$? I don't think you can use the eisenstein criterion here
0
votes
1answer
60 views

Notation meaning:$ R(x^2 + x +1)$

I'm doing a problem sheet on rings and ideals, and would appreciate clarification on some notation used. The problem is: "Let $R=\mathbb{Z}_2[x]$, the ring of polynomials with coefficients in $\...
0
votes
1answer
33 views

$f$ be a ring automorphism on $R[x]$ such that $f(u)=u , \forall u \in R$ , then is it true that $f(x)=ax+b $ for some $a,b \in R$?

Let $R$ be a ring and $f:R[x] \to R[x]$ be a ring automorphism such that $f(u)=u , \forall u \in R$ , then is it true that $f(x)=ax+b $ for some $a,b \in R$ ?
3
votes
2answers
78 views

Existence of Zero Divisors in $C(X,\mathbb{R})$

Consider any topological space $X$ and $\mathbb{R}$ be with usual topology. The set of all continuous functions from $X$ to $\mathbb{R}$, denoted by $C(X,\mathbb{R})$, is a commutative ring with unity ...
0
votes
1answer
39 views

Application of extended euclidean algorithm to find the inverse of polynomial

I'm trying to understand the Example. ( A quotient ring which is a field) on this page:http://www.millersville.edu/~bikenaga/abstract-algebra-1/quotient-rings-of-polynomial-rings/quotient-rings-of-...
1
vote
1answer
94 views

Basic open sets in the Zariski topology are also compact.

Let $A$ be a commutative ring and $X = \text{Spec}(A)$. The closed sets are those of the form $V(E) = \{$ prime ideals $\hat{p} \subset A $ containing $E \}$. And the open sets are the complements ...
2
votes
1answer
161 views

$1+ab$ is a unit if and only if $1+ba$ is a unit. [duplicate]

Let $R$ be a ring with identity and $a,b \in R$ then prove that $1+ab$ is a unit if and only if $1+ba$ is a unit and find the inverse. Then there exist an element say $s \in R$ such that $(1+ab)s =1$ ...
1
vote
1answer
48 views

What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ?

What is the dimension of $\mathbb R[x] / \langle x^3-x\rangle$ as a vector space over $\mathbb R$ ? Can someone please give some links , articles where I can study about polynomila rings and its ...