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
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3answers
133 views

How to prove the one-variable calculus definition of derivative extends to $\Bbb C$ *only* because $\Bbb C$ is a field?

I have been told the one-variable calculus definition of derivative extends to $\Bbb C$ only because $\Bbb C$ is a field. See : Higher dimensional analogues of the argument principle? $$ ...
4
votes
2answers
164 views

Is $2\mathbb{Z}/4\mathbb{Z}$ NOT a field?!

According to wikipedia: If $R$ is a unital commutative ring with an ideal $m$, then $k = R/m$ is a field if and only if $m$ is a maximal ideal. In that case, $R/m$ is known as the residue field. ...
3
votes
7answers
403 views

What's the rationale for requiring that a field be a $\boldsymbol{non}$-$\boldsymbol{trivial}$ ring?

The title pretty much says it all. Of course, one answer (IMO unsatisfactory) to such questions goes something like "a definition is a definition, period." In my experience, mathematical definitions ...
1
vote
1answer
153 views

When is a vector space (over field $K$) also a ring (with subring $K$)?

(Apologies in advance for the very naive question. I'm just learning about all this. Also, for the sake of expedience, below I use the word "ring" when it would more correct for me to use ...
3
votes
0answers
71 views

Graded Betti Numbers of a Graded Ideal with Linear Quotients

Exercise 8.8(a) in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a graded ideal which has linear quotients with respect to a homogeneous system of generators ...
2
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0answers
77 views

Graded Betti Numbers of a Stable Monomial Ideal

Exercise 8.8 in Monomial Ideals by Herzog and Hibi: Let $I\subset S=K[x_{1},...,x_{n}]$ be a stable monomial ideal with $G(I)=\{u_{1},...,u_{m}\}$ and such that for $i<j$, either ...
0
votes
2answers
109 views

How to show that $x$ becomes a root of $p(x)$ in $F[x]/(p(x))$

$F$ is a field, $p(x)$ is irreducible polynomial at $F[X]$. $K=F[X]/\left<p(x)\right>$. For every $a\in F$ we will mark: $\bar{a}=\left<p(x)\right>+a$. Now, the question is: How do I show ...
1
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2answers
193 views

The proof that a finite field has a prime power order

I don't seem to grasp the proof. First we construct a vector space over a subfield with prime order $p$ where $p$ is the characteristic of the field . As the field is finite , the vector space will be ...
0
votes
2answers
104 views

How to find a polynomial product that give me $x^6+1$

I need to find a polynomials product that give me $x^6+1$ at $\mathbb{R}[X]$ and at $\mathbb{C}[X]$. I need that the product will be of irreducible polynumials... Thank you!
0
votes
4answers
69 views

Consider $S = \{(n,n)\mid n\in \mathbb{Q}\}$. Prove $S$ is a subring of $\Bbb Q \times \Bbb Q$ but not an ideal

Consider $S = \{(n,n)\mid n\in \mathbb{Q}\}$. Prove $S$ is a subring of $R$ but not an ideal in $R$, where $R = \mathbb{Q} \times \mathbb{Q}$. I don't know how to do this one... any help would be ...
7
votes
1answer
202 views

Minimal counterexamples of the isomorphism problem for integral group rings

The isomorphism problem for integral group rings asks if two finite groups $G,H$ are isomorphic when their integral group rings $\mathbb{Z}[G]$, $\mathbb{Z}[H]$ are isomorphic. Quite a lot has been ...
0
votes
1answer
86 views

coprime elements

Let $R$ be a ring, then two elements $I,J$ are coprime, if $RJ+RI=R$ or in other words, if there exist $r_1,r_2 \in R$ such that $r_1I+r_2J=u$, where $u$ is a unitity in $R$. Now let $\mathbb{Q}$ be ...
5
votes
3answers
153 views

Can Boolean ring without unit be embedded into a boolean ring?

While going through a book (Lectures on Boolean algebra, Halmos) I got struck at the following question : Prove that every Boolean ring without a unit can be embedded in a Boolean ring with a unit. ...
2
votes
2answers
74 views

Non-unital commutative ring without non-prime ideals?

Does there exist a non-unital commutative ring such that all its proper ideals are prime? Note also that that equipping the abelian group $\mathbb Z/p\mathbb Z$ with trivial multiplication $xy=0$ for ...
2
votes
2answers
58 views

Ideals of a subring of $M_{2\times 2}(\mathbb{R})$

Define $$A:=\left[\begin{array}{cc}\mathbb{R}&\mathbb{R}\\ 0&\mathbb{R}\end{array}\right]=\left\{\left[\begin{array}{cc}a&b\\ 0&c\end{array}\right]:a,b,c\in\mathbb{R}\right\}$$ Prove ...
1
vote
1answer
145 views

Proving that set is (or is not) a field

Let $P = \{a + b\sqrt[3]3 + c\sqrt[3]9, a, b, c \in \Bbb Z \}$ It is easy to prove that $(P, +, \cdot)$ is a ring considering ordinary addition and multiplication. How to prove that this set is or is ...
1
vote
0answers
59 views

Hochschild homology with trivial coefficients: how to make $K$ an $M_n(K)$-module

Let $R$ be a ring, $A$ an associative $R$-algebra, and $M$ an $A$-$A$-bimodule. Then the Hochschild homology of $A$ with coefficients in $M$, denoted $HH_\ast(A)$, is the homology of the chain complex ...
3
votes
3answers
470 views

homomorphism image of a maximal ideal of a ring

Let $R$ and $S$ be commutative rings with $1$ and $\phi: R\rightarrow S$ be a surjective ring homomorphism. Then for an arbitrary maximal ideal $I$ of $R$, does $\phi(I)$ have to be maximal in $S$? ...
-2
votes
4answers
128 views

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$.

Let $K$ be a field and $f(x)\in K[x]$. Prove that $K[x]/(f(x))$ is a field if and only if $f(x)$ is irreducible in $K[x]$. How to prove? I really have no idea... Thank you a lot.
0
votes
1answer
102 views

$gcd(a,b)$ in a UFD subring is not a greatest common divisor in the ring

Give a counterexample that $R$ is a unique factorization domain but not a principal ideal domain, $S$ is a ring containing $R$, such that $a,b\in R$, $gcd(a,b)$ in $R$ is not a greatest common ...
1
vote
1answer
47 views

irreducible elements of polynomial rings

Let $p$ be a prime integer. For $x\in\mathbb{Z}$, let $x'$ be the remainder of $x$ when divided by $p$. Let $\sum_{i=0}^{n}a_iX^i\in \mathbb{Z}[X]$ with $p$ does not divide $a_n$ in $\mathbb{Z}$. Then ...
1
vote
3answers
70 views

Minimal polynomial of a finite field

Let $p,n\in\mathbb{N}$ with $p$ prime and $q=p^n$. Let $\mathbb{F}_q$ be the finite field with $q$ elements (unique up to $\cong$), i.e. the $q$-th Galois field. According to this, the extension ...
9
votes
2answers
129 views

$R/I$ is not Noetherian. Prove that $I$ is a prime ideal.

Let $R$ be a commutative ring with $1$ and let $I$ be an ideal of $R$, maximal with respect to the property that $R/I$ is not Noetherian. Prove that $I$ is a prime ideal. I need some hints to ...
1
vote
1answer
18 views

On distributive modules

Is there a ring $R$ (with identity) such that for any right module $M$ the lattice of submodules of $M$ is distributive?
0
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2answers
57 views

Whats avoid from ideal to be principal?

At this question I asked about specific one... But I think that I don't understand the basic: If I have an ideal $I$, what avoid it to be principal? I think that I really need a good ...
3
votes
1answer
126 views

$B$ is a finitely generated module over $A$. Then if $A$ is a Noetherian ring (or Artinian ring), then $B$ is a Noetherian (Artinian) ring.

Let $A$ be a subring of $B$. Suppose $B$ is a finitely generated module over $A$. Prove that if $A$ is a Noetherian ring (or Artinian ring), then $B$ is a Noetherian (Artinian) ring. I am quite ...
3
votes
2answers
282 views

Localization of an integral domain and fields of fractions

Is it true that every localization of an integral domain is isomorphic to a subring of its field of fractions? How are the localizations of an integral domain related to its field of fractions? Is ...
5
votes
1answer
105 views

On maximal submodules of projective modules

I know that any non-zero projective module has a maximal submodule. But is it true that any proper submodule is contained in a maximal submodule !?
2
votes
2answers
22 views

Cyclic modules over local rings

Let $R$ be a commutative local ring. Is it true that all cyclic $R$-modules are indecomposable? If not please give me an example.
2
votes
1answer
130 views

A property of $I$-adic topologies

Let $R$ be a commutative ring with multiplicative identity and $I$ a proper ideal of $R$ such that the intersection of its powers is the zero ideal. It can be shown that if the $I$-adic topology is ...
2
votes
2answers
228 views

The ring of upper triangular matrices as a module over itself

$R$ is taken to be the ring of upper $3 \times 3$ matrices with entries in $\mathbb{R}$. If I view $R$ as a module over itself, are any of its submodules free? And how can I prove that its ...
2
votes
1answer
134 views

Find all homomorphisms from a quotient polynomial ring $\mathbb{Z}[X] /(15X^2+10X-2)$ to $\mathbb{Z}_7$

I'm completely lost, what my problem is I don't get the gist of a quotient polynomial ring nor ANY homomorphisms between it and some $\mathbb{Z}_n$, much less ALL of them. I know there is something ...
2
votes
2answers
50 views

Finite sum of right ideals is finite sum of left ideals

Let $R$ be a ring with $1$ and let $R=I_1\oplus I_2 \cdots \oplus I_n$ where $I_i$ are right ideals of $R$. I need to show that: i) $I_i=x_iR$ for some $x_i\in I_i$ ii) $x_i^2=x_i$ and $x_ix_j=0$ ...
7
votes
3answers
676 views

Integral domain whose irreducible elements are not prime

Is there some integral domain such that none of its irreducible elements is prime? Recall that a nonzero, non invertible element $a$ of an integral domain $D$ is said to be Irreducible, if for ...
2
votes
2answers
310 views

Should the sum of zero divisors also a zero divisor?

In a general ring $A$ (commutative with $1$), should the sum of two zero divisors also a zero divisor? Could anyone give a proof or a countexample? Moreover, consider the polynomial ring $A[x]$, ...
1
vote
1answer
69 views

Given a ring $R$, when one talks of $R[X]$ what do they exactly mean?

Given a ring $R$, when one talks of $R[X]$ what do they exactly mean? To elaborate, do they mean ring of all elements of the form $a_nX^n + \ldots a_0$ where NO equivalences are present? I.e $a_nX^n ...
0
votes
1answer
45 views

products in a unique factorisation domain.

I'm learning a proof about UFD's but I'm not following this particular part: Given a ring A which is a UFD, with a non-zero non-unit element p which is a prime element. Then if you take an arbitrary ...
1
vote
2answers
126 views

Calculations in quotient rings

Is there a nice way to perform calculations within a quotient ring? for instance, an exercise I had was to show $\mathbb{Z}[X]/(X^2+1) \cong \mathbb{Z}[i]$, but this got me thinking; If I didn't know ...
1
vote
1answer
79 views

On a particular $K[x,y]$-module

This is a follow up from HERE. Suppose $K$ is a field and consider $K$ as a $K[x,y]-$module where the scalar product is defined by $f(x,y)\cdot k = f(0,0)\cdot k$. Is $K$ injective or flat as ...
4
votes
2answers
152 views

Is an ideal which is maximal with respect to the property that it consists of zero divisors necessarily prime?

This is in follow-up to this question. Let $R$ be a commutative ring with identity and consider the set $Z \subset R$ of zero divisors. If the ideal $I\subset Z$ is maximal with respect to the ...
8
votes
2answers
328 views

Can the product of two non invertible elements in a ring be invertible?

Let $A$ be a unitary ring. The question is simply: can the product of two non invertible elements in $A$ be invertible? I proved that the answer is negative if $A$ does not have zero divisors, ...
16
votes
7answers
841 views

Introduction to ring theory?

I've been teaching myself algebra these couple of months. I already went through the basics of group (Lagrange, action, class equation, Cauchy and Sylow theorems etc.) And I already have some linear ...
0
votes
1answer
77 views

Fundamental questions on rings of polynomials.

Put $\mathfrak{E}$ the union of $(0,0)$ and $k\times 1$ in $k^2$ ($k$ an algebraically closed field). Furthermore let $\mathfrak{Z}$ the set of all $f\in k[x,y]$ such that $f(s)=0$ for all ...
3
votes
1answer
156 views

to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal

im asked to prove that the set consisting of all zero divisors in a commutative ring with unity contains at least one prime ideal. i cant even start in the proof , ive just defined my set but cant ...
1
vote
2answers
103 views

How to construct finite fields of any prime power order?

For a prime $p$, I know that $\mathbb Z_p$ is a field. To construct a field with four elements, I know I can just take $\frac{\mathbb Z_2[x]}{(x^2+x+1)}$. Similarly, to construct a field of order ...
0
votes
1answer
44 views

the direct image of an ideal needs not to be an ideal

I need an example of a ring mapping homo such that the image of an ideal needs not to be an ideal ? I found that the image is an ideal if the mapping was an onto one ! so all we need to find a mapping ...
1
vote
2answers
119 views

Prove or disprove that the ideal $(2+4\mathbb{Z},x)$ is a principal ideal in $(\mathbb{Z}/4\mathbb{Z})[x]$

Prove or disprove that the ideal $(2+4\mathbb{Z},x)$ is a principal ideal in $(\mathbb{Z}/4\mathbb{Z})[x]$. I know $\mathbb{Z}/4\mathbb{Z}$ is not a field. have something help?
1
vote
1answer
48 views

Ring of Polynomials with two variables

I am working on the ring $R$ of polynomials with two variables over some field, and I have the ideal $J$ generated by $x^2$ and $xy$. I am considering the module $R/J$ over $R$; and I get to a ...
3
votes
1answer
92 views

Suppose that $I$ is an ideal of $R$ which is maximal with respect to the property that it is proper and not prime.

Let $R$ be a commutative ring with $1$. Suppose that $I$ is an ideal of $R$ which is maximal with respect to the property that it is proper and not prime. Deduce that $I$ is contained in at most ...
3
votes
1answer
136 views

Ideals containing $(6, x^3-1)$ in $\mathbb{Z}[x]$

I want to find all ideals containing $(6, x^3-1)$ in $\mathbb{Z}[x]$ and I can only find ten ideals: $\mathbb{Z}[x]$ $(2, x-1),\; (2, x^2+x+1),\; (3, x-1)$ $(6,x-1),\; (2,x^3-1),\; (6, x^2+1+1),\; ...