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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2
votes
1answer
39 views

Basis of a free module is the localization of its basis?

Let $R$ be a ring and $M$ a free $R$-module with basis $B$. If $P$ is a prime ideal of $R$, is $B_P$ a basis for the $R_P$-module $M_P$?
2
votes
0answers
36 views

Finding the general form of an element in $\frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$

I'm trying to find the general form of elements in the quotient ring: $$R = \frac{\mathbb{Z}_4 [x]}{(x^2 + 1)}$$ Now my initial thoughts are to take a general element $f \in R$ so that $f = g + (x^2 ...
2
votes
1answer
89 views

The family of schemes $\operatorname{Spec} A[x]/(x^n)$

Consider the family $S_n:=\operatorname{Spec} A[x]/(x^n)$ of schemes, $A$ denoting any ring (which in our subject always means commutative and with identity). Is there some intuitive picture for ...
1
vote
2answers
47 views

roots of cubic polynomial

On page 26 of Milne's Elliptic Curves (http://www.jmilne.org/math/Books/ectext5.pdf), he states the following: "... a cubic polynomial $h(x) \in k[x]$ with two roots in $k$ has all of its roots in ...
2
votes
1answer
29 views

Let $R$ a ring with maximum common divisor. If $a,b,c \in R$ such that $a|bc$ and $(a,b)=1$ then $a|c$.

Let $R$ a ring with maximum common divisor. Show that if $a,b,c \in R$ such that $a|bc$ and $(a,b)=1$ then $a|c$. Comments: I tried to use the Bezout's theorem, but in my course we saw it only ...
1
vote
0answers
24 views

Why is $A/I^n A \cong \hat{A}/I^n \hat{A}$?

Let $A$ be a commutative ring, $I \subset A$ a finitely generated ideal. Define $\hat{A} := \varprojlim A/I^n$. What is the best way to proof that $A/I^nA \cong \hat{A}/I^n \hat{A}$ for all $n$?
0
votes
2answers
30 views

Question related to integrality of field of fractions

This is actually not a problem, but it's a statement which is taken for granted and I don't know how to prove it. Hope some one can help me. I really appreciate: Suppose $A$ is subring of ...
2
votes
1answer
42 views

Prove that $\beta$ is a root of a $n$ degree polynomial of K [x] with leading coefficient $1$.

Prove that $\beta$ is a root of a $n$ degree polynomial of K [x] with leading coefficient $1$. Here $K$ is UFD, $Q$ is its field of fractions, $Q(\beta)$ is algebraic extension of $n>1$ degree of ...
1
vote
1answer
34 views

Proving $M_p$ is maximal in $C[0,1]$

Let $M_p$ be the ideal of those continuous functions of $C[0,1]$ which have $p\in [0,1]$ as a zero. It is a commonly known fact that $M_p$ is a maximal ideal. However, the proof is generally ...
1
vote
0answers
33 views

Complex matrices: looking for homomorphism

Let $\mathbb{C}$ denote the complex numbers, and let $M_2(\mathbb{R})$ be the ring of $2$ by $2$ matrices with real entries. Define a function $f:\mathbb{C} \to M_2(\mathbb{R})$ by $ f(a+bi) = ...
0
votes
1answer
43 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
votes
2answers
45 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
145 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
27 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: ...
1
vote
1answer
109 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 ...
2
votes
1answer
65 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
1
vote
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 ...
3
votes
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
votes
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 ...
-1
votes
1answer
42 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
votes
1answer
89 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
115 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
58 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
votes
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
118 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
vote
1answer
91 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
votes
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
votes
1answer
83 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 ...
1
vote
2answers
47 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
vote
1answer
47 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
votes
2answers
81 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
vote
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
vote
0answers
19 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, ...
5
votes
2answers
96 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
44 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
vote
0answers
42 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
74 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
37 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 $Spec(R') \to Spec(R)$ , ...
1
vote
1answer
43 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
vote
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
vote
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
164 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
60 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 ...
-2
votes
2answers
50 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 ...
1
vote
3answers
128 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 ...
4
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
votes
1answer
39 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
32 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
votes
1answer
67 views

extension of Noetherian rings

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