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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1answer
44 views

Prove $R \times R$ is NOT an integral domain

I have a question, and this is it in entirety: Let R be a non-zero ring(that is, R contains at least one element other than the zero element). Prove that $R \times R$ is NOT an integral domain. ...
3
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2answers
46 views

ACC on principal ideals implies factorization into irreducibles. Does $R$ have to be a domain?

I am following an argument in chapter zero of Eisenbud's Commutative Algebra book. It is not clear whether or not he is assuming that $R$ is a domain. If I start the proof assuming $R$ is not ...
9
votes
2answers
152 views

What kind of object is the kernel of a ring homomorphism?

The category $\mathbf{Grp}$ of groups has a zero object, namely the trivial group $1$. Since $\mathbf{Grp}$ is furthermore complete, we have the notion of a kernel of a group homomorphism. The kernel ...
1
vote
1answer
27 views

On the existence of polynomial roots

Assume $F$ is a field, and $f\in F[x]$ is polynomial. To see that $f$ has a root in some extension of $F$, without loss of generality we can assume $f$ is irreducible. Indeed any polynomial $f$ is ...
0
votes
1answer
29 views

Showing the following rings are isomorphic

Is it true that $\frac{\mathbb{Z}[x]}{(3,x^6+1)}\cong{\frac{\mathbb{Z_{3}[x]}}{(x^6+1)}}$? I believe that it is but am not sure how to justify it. The idea I want to use is this: ...
0
votes
1answer
110 views

Subset of $\mathbb{Z} \times \mathbb{Z}$

I have a past exam question that is as follows: Let $k$ be a fixed integer and $S = \{(a,ka)|a \in \mathbb{Z}\}$ be a subset of $\mathbb{Z} \times \mathbb{Z}$. Prove that $S$ is a subgroup of ...
1
vote
1answer
66 views

Show that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic to the field $\mathbb{Q}$

I have problem in showing that $\operatorname{End}_{\mathbb{Z}}(\mathbb{Q})$ is isomorphic as a ring to the field $\mathbb{Q}$. Any idea? Thanks
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0answers
20 views

Monotonic Union of Submodule

The theorem is: Suppose $M$ is an $R$-module, $T$ is an index set, and for each $t \in T$, $N_t$ is a submodule of $M$. If $\{N_t\}$ is a monotonic collection, $\cup_{t\in T} N_t$ is a submodule. Is ...
2
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0answers
24 views

Need help with finding generator

$I=\{a+bi \in R\mid a \equiv b\pmod 2\}$ is ideal of $R=Z[i]=\{a+bi\mid a,b \in Z\}$. Can somebody help me to find the generator of $I$?
0
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0answers
28 views

Proving a left ideal in a direct sum is also a right ideal.

Assume that $R$ is a ring with $1$ such that every $R$ module is injective. I've proved that $R$ is the finite direct sum of left ideals. I want to write $R$ as a ring direct product of $2$-sided ...
0
votes
1answer
44 views

Ring of linear transformations modulo finite rank transformations [closed]

Let $ K $ be a field and $ V $ be a vector space of countable dimension (infinite) over $ K $, and let $ L = L (V) $ be the vector space of $ K $-linear transformations on $ V $. Let $ I $ be the ...
5
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1answer
92 views

On the indecomposable decomposition of the reduction of an integral representation

Here is a problem I have been grappling with all day. I started out thinking it might be true but am now inclined to believe it is false, and would like to see a counterexample. Suppose $G$ is a ...
2
votes
3answers
118 views

$S^{-1}A \cong A[x]/(1-ax)$ [duplicate]

If $A$ is a commutative ring with unit, $a \in A $ and $S = \lbrace a^n \mid n \geq 0 \rbrace $ then there is an isomorphism $$S^{-1}A \cong A[x]/(1-ax).$$ In fact we can consider the ...
0
votes
1answer
60 views

If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ satisfies $S_1$

Let $I$ be an ideal of polynomial ring $R=K[x_1,\ldots,x_n]$ and $x$ be a non-zero divisor of $R/I$. Is the following statement true? If $R/I$ satisfies Serre's condition $S_2$ then $R/(I,x)$ ...
0
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0answers
60 views

General Technique for Finding all Ring Homomorphisms Between Two Rings

I'm seeing a bunch of questions such as "find all ring homomorphisms from $\mathbb{Z}$ to $\mathbb{Z}/30\mathbb{Z}$," etc., lately, so I'm wondering about whether there's a general method for doing ...
2
votes
2answers
124 views

Applications of groups, rings and modules in life [closed]

I have read calculus. Now I'm reading algebra. What is the application of groups, rings and modules in life? When we study derivations we know several applications. What about groups, rings and ...
1
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1answer
92 views

Morphism of rings and localization

Let $ \varphi : A \to B $ be a morphism of rings. Why are the two following assertions equivalent: $ 1) $ There exists a multiplicative subset $ S $ of the ring $ A $, and an ideal $ I $ of $ A $, ...
3
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1answer
47 views

A local PID is a Euclidean domain

Studying commutative algebra I've encountered this statement: A PID which is also a local ring is a Euclidean domain. Is it true ? Why ?
0
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1answer
84 views

Studying $\operatorname{Spec}\mathbb{Z}[x]$, $\operatorname{Spec}\mathbb{R}[x]$, and $\operatorname{Spec}\mathbb{C}[x,y]$.

While there is a similar question here but that was marked as a duplicate to this question. The latter question, at the level that I am at doesn't give me much insight. I also thought that if I could ...
0
votes
2answers
48 views

Integral domain (rings and fields)

Let $\mathbb Z[i]=\{a+ib \mid a, b \in \mathbb Z \}$. How to Show that $\mathbb Z[i]$ is a integral domain?
1
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1answer
48 views

Commutative ring can be homomorphically mapped onto field

During my algebra lecture, my lecturer used the fact that any commutative ring can be homomorphically mapped onto a field. Is the statement true? How to show that? Thanks
2
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2answers
82 views

Why is $ \mathrm{Frac} ( A / \mathfrak{p} ) = A_{\mathfrak{p}} / \mathfrak{p} A_{\mathfrak{p}} $? [duplicate]

$ A $ is a commutative ring, $ \mathfrak{p} \in \mathrm{Spec} A $, $ A_{\mathfrak{p}} = ( A \backslash \mathfrak{p} )^{-1} A $, $ \mathrm{Frac} ( A / \mathfrak{p} )$ is the field of fractions of $ A / ...
1
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1answer
34 views

The group morphism of tye ring

Let $(G,+,\cdot)$ $(H,+,\cdot)$ be rings, we suppose that the unites $(G*,\cdot)$ and $(H*,\cdot)$ form groups respectively, for example, the matrix ring $M(n,\mathbb{R})$. There is a group morphism ...
1
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0answers
20 views

Why is the norm of an ideal contained in that ideal?

Suppose $K$ is a number field and that $\mathcal{O}_K$ is the ring of integers of $K$. Now, let $I$ be an ideal in $\mathcal{O}_K$. I know that $N(I) \in I$, but I want to prove it. By definition, ...
4
votes
2answers
98 views

The prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $.

I have to study the prime elements of the ring $ \mathbb{Z} \! \left[ \dfrac{i \sqrt{3} - 1}{2} \right] $. For the moment, I cannot find the general form of such elements. Can you help me? Thanks! :) ...
0
votes
1answer
48 views

Computing a regular sequence of generators for an ideal

Let $R = \mathbb{C}[x_1,\ldots,x_n]$. Let $I$ be an ideal, and suppose we know a finite list of generators for $I$, say $I = \langle f_1,\ldots,f_k\rangle$. Is this information enough to compute a ...
1
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0answers
44 views

Finitely generated idempotent ideal must be generated by an idempotent [duplicate]

Let $A$ be a commutative but not necessarily unital ring. How can we show that a finitely-generated ideal $I$ of a ring $A$ satisfying $I=I^2$ is generated by an idempotent element?
0
votes
4answers
78 views

Maximal ideal in the ring of polynomials over $\mathbb Z$

Let $\mathbb Z[x]$ the ring of polynomials with integers coefficients in one variable and $I =\langle 5,x^2 + 2\rangle$, how can I prove that $I$ is maximal ideal. I tried first see that $5$ and ...
0
votes
1answer
39 views

Relation between faithfully flatness and map of $Spec$

I'm stuck on this exercise ( from Bosch ) : Let $\phi :R \to R' $ a flat ring morphism. Show that $\phi$ is faithfully flat if and only if the associated map $Spec(R') \to Spec(R)$ , ...
0
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1answer
46 views

Ideals problem from Hungerford Algebra

This question is from Hungerford's Algebra. Let $R$ be a ring with identity and $S$ the ring of all $n\times n$ matrices over $R$. $J$ is an ideal of $S$ iff $J$ is the ring of all $n\times n$ ...
1
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1answer
28 views

Canonical homomorphisms $R_{\mathfrak{p}_i} \to R/\mathfrak{p}_i^n$ are isomorphisms when $R$ is artinian

I'm doing this exercise (from the book of Bosch): Let $R$ be an Artinian ring and let $\mathfrak{p}_1, \ldots \mathfrak{p}_n $ be its (pairwise different) prime ideals. Show that: a) The ...
1
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1answer
51 views

Rings which are finitely generated and free over Cohen-Macaulay rings are also Cohen-Macaulay

Let $S$ be a Cohen-Macaulay (C-M) ring, and $R$ a ring containing $S$ such that as an $S$-module is finitely generated free. Could we deduce that $R$ is also C-M? I guess probably we could use ...
7
votes
1answer
166 views

Pseudo associated primes and short exact sequences

Let $A$ be a commutative ring, and $$0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow0$$ a short exact sequence of $A$-modules. The following inclusion relation is well-known: ...
0
votes
2answers
62 views

Prove that $6$ and $2(1+\sqrt{-5})$ do not have a gcd in $\mathbb{Z}[\sqrt{-5}]$ [closed]

Prove that $6$ and $2(1+\sqrt{-5})$ do not have a gcd. here those elements belong to $\mathbb{Z}[\sqrt{-5}]$. They have common divisors like the number $2$ But if $a$ is another divisor it must ...
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2answers
51 views

If $p$ is an irreducible element of an integral domain $D$, and if $e$ belongs to $D^\ast$ prove that $ep$ is also ireducible

If $p$ is an irreducible element of an integral domain $D$, and if $e$ belongs to $D^\ast$, prove that $ep$ is also ireducible . To me it seems so profound, but I cannot get the proof . I'm trying to ...
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3answers
136 views

Questions about $\Bbb Q[\sqrt{p}]$ and $\Bbb Q(\sqrt{p})$

I studied this part where they talk about $\Bbb{Q}(\sqrt{2})$ and $\Bbb{Q}[\sqrt{2}]$ and I really start to get confused. Definitions: $$ \Bbb{Q}[\sqrt{2}] = \left\{ a + b \sqrt{2} \mid a,b \in ...
3
votes
5answers
81 views

Show that $f^{-1}(\langle0\rangle)$ is not a maximal ideal of $\mathbb{Z}$.

Let $f\colon \mathbb{Z} \to \mathbb{Q}$ be a ring homomorphism. Show that $f^{-1}(\langle0\rangle)$ is not a maximal ideal of $\mathbb{Z}$.
0
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1answer
43 views

About second uniqueness primary decomposition theorem

I'm self-learning commutative algebra from Introduction To Commutative Algrebra of Atiyah and Macdonald and get frustrated about the second uniqueness primary decomposition theorem. I copy the theorem ...
0
votes
1answer
38 views

When a monomial ideal is primary

I know that a monomial ideal in $k[x_1, \ldots x_n]$ with $k$ a field is prime if and only if is of the following type $$I = (x_{i_1}, \ldots \ ,x_{i_k})$$ Is there a similar criterion to establish ...
2
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1answer
97 views

Extension of Noetherian rings [closed]

Let $A \subset B$ rings. If $B$ is noetherian then $A$ is noetherian. Is false or true?
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1answer
52 views

Height of finitely generated ideals in a catenary local ring

If $R$ is a noetherian local domain which is catenary, and $a_1,...,a_n$ are elements of the maximal ideal of $R$ with $\operatorname{height}(a_1,...,a_n)=n$, could we conclude that ...
-1
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2answers
43 views

Ring homomorphisms map non units to non units

A ring homomorphism maps units to units. I was wondering if it implies that it maps non units to non units. I tried to find a counter example because I think the answer should be no but couldn't find ...
2
votes
1answer
90 views

Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian?

Question: Is the ring $ R = \{ f \in \mathbb{C}[x,y] \mid {\nabla f}(0,0) = (0,0) \} $ Noetherian? I guess it isn’t Noetherian as I suspect that $$ (x y + y^{2}), \quad (x y + y^{2},x^{2} y + ...
1
vote
1answer
81 views

Finitely many prime ideals $\Rightarrow$ cartesian product of local rings

I'm stuck on this problem from Bosch, Algebraic geometry and commutative algebra: Let $R$ be a commutative ring containing only finitely many prime ideals and assume that a certain power of the ...
1
vote
0answers
42 views

Rings with noncommutative addition

I was wondering if "rings" with noncommutative addition are studied at all? Of course, if a ring $R$ has a $1$, then for all $a, b\in R$, $a+a+b+b=(1+1)a+(1+1)b=(1+1)(a+b)=(a+b)+(a+b)=a+b+a+b$, from ...
1
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1answer
25 views

Show that if $M$ is a R-module free, for all $r \in R$ and $m \in M$ such that $r \neq 0$ e $m \neq 0$ we have $rm \neq 0$.

is it true that if $M$ is a $R$-module free, for all $r \in R$ and $m \in M$ such that $r \neq 0$ e $m \neq 0$ we have $rm \neq 0$. Comments: I tried to do the following: Suppose that $rm = ...
6
votes
3answers
106 views

Multiplicative inverse of $0$

If I'm not mistaken, in a ring with identity, the additive identity cannot have a multiplicative inverse. I'm trying to prove this. Here's my attempt so far: Suppose $0\cdot a=1$ $$0\cdot a=1$$ ...
1
vote
1answer
56 views

Ring extension and Jacobson rings

If $R\subseteq S$ are commutative rings, is it a fact that $R$ is a Jacobson ring if and only if $S$ is so? I guess the contraction of maximal and prime ideals of $S$ may be helpful in this ...
2
votes
3answers
67 views

Must an ideal generated by an irreducible element be a maximal ideal?

If you have an irreducible element say $b$ in a ring, is the ideal $\langle b\rangle$ a maximal ideal?
0
votes
1answer
22 views

Let $R$ be a UFD and $p,q,r \in R$. $pq=r^3$ and $\gcd(p,q)=1$ then $p,q$ are cubes up to associates.

I'm not too sure how to prove this statement. This seems like a relatively small problem however I can't for the life of me figure out how to start this so I don't really have any working to show. I ...