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
61 views

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field?

If $F$ be a field, then $F[x]$ is a principal ideal domain. Does $F$ have to be necessarily a field? My Thoughts: Suppose instead of $F$, we take the set of polynomials $R[x]$ over a commutative ring ...
3
votes
2answers
41 views

A doubt about lower nil radical while proving 2-primality of ring.( Baer-McCoy Radical)

I was proving that a reversible ring is 2-Primal for an exercise in T.Y Lam's book, but I got stuck. Here is where I'm stuck: let $a$ be a nilpotent element of $R$ with $a^n=0$. Then using ...
0
votes
1answer
36 views

Global dimension regular rings of finite type

Have I made an error in my reasoning? If $k$ is a field, $A$ is a commutative regular $k$-algebra of finite type and ${\mathfrak{m}}$ is a maximal ideal in $A$ then since $Ext_{A_{\mathfrak{m}} ...
0
votes
2answers
41 views

A ring with prime characteristic

Let $p$ be a prime and let $R$ be a commutative ring with characteristic $p$. Prove that the number of elements of the set $$S_k=\{x\in R\;\lvert \;x^p=k\}\quad \text{for} \quad k\in ...
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votes
0answers
43 views

Is every local ring the localization of some other ring?

One way of constructing a local ring is to start with any commutative ring, and localize all the elements outside of some maximal ideal (i.e., adjoining inverses to all those elements). But I'm ...
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3answers
65 views

Prove that a polynomial of degree $n$ over a commutative ring with zero divisors may have more than $n$ zeroes?

A polynomial of degree $n$ over a commutative ring with zero divisors may have more than $n$ zeroes. Attempt: Let $R$ be the commutatve ring which has a zero divisor $a \neq 0$. Then $\exists~~b \in ...
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2answers
42 views

prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.

let $R$ be a unique factorization domain and let $F$ be its field of fractions. Prove that if $p(x)\in R[x]$ is reducible over $F[x]$ then $p(x)$ is reducible over $R[x]$.
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1answer
41 views

Show that if M, N are non-zero commutative rings, then M×N always has zero divisors, and is not an integral domain or a field.

Show that if M, N are non-zero commutative rings, then M×N always has zero divisors, and is not an integral domain or a field. How do I do this?!
5
votes
1answer
63 views

Irreducibility of some multivariate polynomials

Consider the polynomials $xw-yz\in A[x,y,z,w]$ and $x^n+y^n+z^n\in A[x,y,z]$, where $A$ is a commutative ring. I am curious to know what conditions on $A$ (factorial ring, algebraically closed field, ...
1
vote
2answers
44 views

Prove that the field of quotients of an integral domain $D$ is the smallest field containing $D$. . My Attempt Shown

Let $D$ be an integral domain and let $F$ be the field of quotients of $D$. Show that if $E$ is any field that contains $D$, then, $E$ contains a sub field that is ring isomorphic to $F$. Hence, the ...
0
votes
1answer
32 views

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism

Suppose that $n$ divides $m$ and that $a$ is an idempotent of $Z_n$. Show that the mapping $x \rightarrow ax$ is a ring homomorphism Attempt: Let $\Phi: Z_m \rightarrow Z_n$ be a ring homomorphism ...
1
vote
2answers
36 views

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. My Attempt Shown

Let $F$ be a field. Show that the field of quotients of $F$ is ring isomorphic to $F$. Attempt: Let $F'$ be the field of Quotients of the field $F$. Let $\Phi:F \rightarrow F'$ such that ...
0
votes
1answer
49 views

Finitely generated ring.

Let $R$ be a finitely generated ring. Then is it true that $R^2$ is also finitely generated? My Attempt: I do not find a counterexample. I think it is true. Please someone help me to prove that ...
4
votes
1answer
69 views

What is necessary and/or sufficient for polynomials to provide isomorphic quotientrings?

Let $R$ be a commutative ring (with identity). Let $f,g\in R\left[x\right]$ both be monic polynomials of degree $d$. Then the underlying abelian groups of the rings ...
0
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2answers
42 views

$R^{(I)} \cong K \oplus H$ where $R^{(I)}$ is free but $K$ is not free

Let $R$ be a commutative ring with unit. Is there an example of a direct sum of $R$-modules $$R^{(I)} \cong K \oplus H$$ where $R^{(I)}$ is free but $K$ is not free ? Clearly $R$ can't be a PID.
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0answers
21 views

Classification of separable algebras over a commutative ring

A separable algebra over a field can be classified as a finite product of matrix algebras over division algebras whose centers are finite dimensional separable field extensions of the field. (See ...
0
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1answer
19 views

an ideal of matrix ring which is projective

Let $K$ be a field and $$ A=\left\{ \begin{pmatrix} a&b&c\\ d&e&f\\ 0&0&g \end{pmatrix} :a,\dots,g\in K \right\}, $$ then $$ J=\left\{ \begin{pmatrix} 0&0&c\\ ...
1
vote
2answers
40 views

The $i^{th}$ prime in a given ring R

When I say that $p_1=2$, I mean that the first prime in the standard ring of integers $(\mathbb{Z},*,+)$ is $2$. I was wondering whether the notion of ordering the primes like this can be generalized ...
2
votes
1answer
42 views

Kahler forms of a smooth affine algebra vanish eventually?

If $k$ is a Noetherian ring, then do the Kahler forms of a smooth affine $k$-algebra of dimension $d$ vanish above $d$? I mean is: $\Omega^{d+1}_{A|k}\cong 0$?
4
votes
1answer
97 views

$z\in\mathfrak R$ iff for every $a\in A$ there is $w$ for which $z+w=zaw=waz$.

In his BAII, Jacobson gives the following exercise, which he attributes to McCrimmon. Show that $z\in\mathfrak R(A)$ iff for each $a\in A$ there exist $w\in A$ such that $z+w=zaw=waz$. I have ...
0
votes
0answers
41 views

Units and Primes in a Ring

Is it true that units in a ring (maybe involves in quaternions) have norm of 1? (norm of 1 does not imply that it is a unit, right?) What about the statement that the number is prime if and only if it ...
4
votes
1answer
149 views

Direct product of finitely many Noetherian non-unital rings is Noetherian

Let $A_1, A_2,...,A_n$ be Noetherian rings (not necessarily unital). Is the direct product $A:=A_1×A_2×⋯×A_n$ necessarily a Noetherian ring? If $A_1, A_2,...,A_n$ are unital, then one can prove ...
3
votes
4answers
112 views

A field with characteristic $0$ contains $\mathbb Q$

To prove that a field $F$ with characteristic $0$ contains $\mathbb Q$, the following lemma is used. Lemma: Let $R$ b a ring with unity. If the characteristic of $R$ is $0$, then $R$ contains a ...
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vote
0answers
82 views

What is $\operatorname{Ass}\operatorname{Ext}^i(M,N)$?

This is exercise 1.2.27 of Bruns-Herzog: Let $R$ be a Noetherian ring, $M$ a finite $R$-module and $N$ an arbitrary $R$-module. Deduce that $\operatorname{Ass}(\operatorname{Hom}_R(M,N)) = ...
0
votes
2answers
80 views

Number of elements and units in quotients of $\mathbb Z[i]$

I was given this problem but have no idea how to solve it. How many elements are there in $\Bbb Z[i]/(3)$? In $\left(\Bbb Z[i]\right)/(3+2i)$? In $\Bbb Z[i]/(5)$? How many units are there in each of ...
4
votes
2answers
41 views

Why is an extension $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism if $\phi : F[x] \to F(a)$ is injective?

Why is $\bar \phi : F(x) \rightarrow F(a)$ an isomorphism given $\phi : F[x] \rightarrow F(a)$ satisfy $\ker \phi = \{0\}$ ? I've been trying to figure out why $\bar \phi$ is an isomorphism, and ...
1
vote
2answers
91 views

Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is an integer $n > 1\mid a^n =a$. Every prime ideal is maximal?

Suppose that R is a commutative ring with unity such that for each $a$ in $R$ there is a positive integer $n > 1$ such that $a^n =a$. Prove that every prime ideal of $R$ is a maximal ideal of R. ...
1
vote
1answer
45 views

Localizations of $ \mathbb{Z}_{p^k}$

Let $S \subseteq \mathbb{Z}_{p^k} $ be a multiplicative subset, where $p$ is a prime number, $k$ an integer. Is it true that $$S^{-1} \mathbb{Z}_{p^k} \cong \mathbb{Z} /n\mathbb{Z} $$ for some ...
2
votes
1answer
33 views

Congruence in localization of rings

Please help me to prove for all maximal ideals $\mathfrak{m}$ of $R$, $(aR/a^2R)_\mathfrak{m}\cong (aR)_\mathfrak{m}/(a^2R)_\mathfrak{m}\cong aR_\mathfrak{m}/a^2R_\mathfrak{m}$, where $R$ is a ...
2
votes
3answers
51 views

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal. What could be wrong in this approach?

In a principal ideal domain, prove that every non trivial prime ideal is a maximal ideal Attempt: Let $R$ be the principal ideal domain. A principal ideal domain $R$ is an integral domain in which ...
0
votes
1answer
37 views

Suppose that $I$ is an ideal of $J$ and that $J$ is an ideal of $R$. prove that if $I$ has a unity then $I$ is an ideal of $R$

Suppose that $I$ is an ideal of $J$ and that $J$ is an ideal of $R$. prove that if $I$ has a unity then $I$ is an ideal of $R$ Attempt: Given that $I$ is an ideal of $J$ which means : ...
2
votes
0answers
71 views

Automorphism of certain f.g. free modules

This is a quick question from Frohlich and Taylor's Algebraic Number Theory, II.4, p 94. Let $R$ be a Dedekind domain with quotient field $K$, $\mathfrak p$ is a non-zero prime ideal of $R$ and ...
2
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0answers
32 views

Hochschild cohomology of skew polynomial rings

Definition The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on ...
2
votes
1answer
43 views

Prime radical that is nil but not nilpotent

Please help me to show that the prime radical of the ring $R=\prod\limits_{n = 1}^\infty { \mathbb{Z} /2^n\mathbb{Z} } $ is nil but not nilpotent.
0
votes
1answer
32 views

Positive structure constants for rings with basis

Suppose $\mathbf{Z}^n$ is endowed with an associative ring structure so that the coefficient of $e_k$ in $e_i e_j$ is a nonnegative integer for every $i,j,k$. Suppose the ring has an identity element ...
1
vote
1answer
32 views

Basis for the completion of a free module

This (or similar) question might have been asked before- apologies for any duplication. I've got a Dedekind domain $R$, a non-zero prime ideal $P$ of $R$ and the completion $\widehat{R}$ of $R$ wrt ...
1
vote
1answer
117 views

Localization and direct limit

Let $ A $ be a ring. Let $ I $ be a preordered set, filtering. Let $ \Sigma $ a multiplicative subset of $ A $. Suppose for any given $ i \in I $ a multiplicative subset $ S_i $ of $ A $ contained ...
2
votes
1answer
48 views

Is the zero of a field irreducible?

Definition1: An element $a$ of an integral domain $R$ is irreducible if it is not a unit and if $a=bc$ implies that $b$ is a unit or $c$ is a unit. Definition2: An element $a$ of an integral domain ...
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votes
1answer
39 views

Ring theory(addition table)

(S,+,.) is a ring , where S={a,b,c,d}. Complete the table. $$ \begin{array}{c|ccccc} + & a & b & c & d \\ \hline a &a &b &c &d \\ b &b &1 &2 &3\\ ...
2
votes
1answer
42 views

In arbitrary commutative rings, what is the accepted definition of “associates”?

In an integral domain, the following are equivalent: $r \mid s$ and $s \mid r$ $r=us$ for some unit $u$ However in arbitrary commutative rings this is no longer the case; in particular, (2) ...
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votes
0answers
19 views

Relation between von Neumann regular rings, Krull dimension 0, and rings with no nonzero nilpotents. [duplicate]

Why a ring $R$ is von Neumann regular if $R$ has no nonzero nilpotents and $\dim R=0$?
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2answers
43 views

Jacobson radical in $A[x]$ where $A$ is a ring.

Let $A$ be a commutative ring, and $A[x]$ be the ring of polynomials in an indeterminate x, with coefficients in A. I found several proofs online that in this case the Jacobson radical equals to the ...
4
votes
1answer
70 views

Help with a problem from Christian Peskine's book about Artinian rings

I am stuck with this problem from the book of Complex Projective Geometry. Let $A$ be a Noetherian ring. Assume that if $a \in A$ is neither invertible nor nilpotent, then there exist $b \in A$ such ...
2
votes
2answers
53 views

Show that $A = \{(3x,y)~|~ x,y \in Z\}$ is a maximal ideal of $Z \oplus Z$. My Attempt Shown

Show that $A = \{(3x,y)~|~ x,y \in Z\}$ is a maximal ideal of $Z \oplus Z$. Here's my attempt, Please tell me where did I go wrong. Attempt: When $R$ is a commutative ring with unity and $I$ is any ...
4
votes
2answers
48 views

Suppose that $R$ is a commutative ring and $|R|=30$. If $I$ is an ideal of $R$ and $|I|=10$, prove that $I$ is maximal ideal

Suppose that $R$ is a commutative ring and $|R|=30$. If $I$ is an ideal of $R$ and $|I|=10$, prove that $I$ is maximal ideal Solution: $|R/I|=3 \implies R/I \approx Z_3$ which is a field. If $R$ is ...
0
votes
1answer
28 views

The nil-radical is an intersection of all prime ideals proof

Every proof I found online made the same implications. Take one for example: http://www.artofproblemsolving.com/Wiki/index.php/Nilradical I'm quoting the relevant part, which confuses me: "To show ...
3
votes
1answer
34 views

If an integral domain $R$ has a factorization basis, is it a UFD?

By a factorization basis for an integral domain $R$, let us mean a subset $\xi$ of the commutative monoid $R^\times = R \setminus \{0\}$ such that firstly, no two elements of $\xi$ are associates, and ...
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1answer
30 views

Constructing an Algebraically Closed Space

How would one construct the simplest -- or canonical, if a canonical construction exists -- countably infinite algebraic closure of the natural numbers?
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2answers
34 views

A question about a ring in the form of a direct product.

Atiyah-Macdonald's "Introduction to Commutative Algebra" (Pg. 20) says the following: Suppose that the ring $A$ is a direct product $\Pi_{i=1}^n A_i$. Then the set of all elements of $A$ of the ...
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0answers
24 views

$ \mathrm{Spec} ( A \times B ) = \mathrm{Spec} A \coprod \mathrm{Spec} B $ [duplicate]

Let $ A $ and $ B $ be two commutative rings. Why is : $ \mathrm{Spec} ( A \times B ) = \mathrm{Spec} A \coprod \mathrm{Spec} B $ ?. Thanks a lot.