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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4
votes
1answer
47 views

Ideal of a product ring?

I am trying to prove whether or not the ideal generated by $\langle (2,2)\rangle$ is a prime ideal of $\mathbb Z_4\times \mathbb Z_4$? My issue is I'm not sure how to do the coordinate ...
1
vote
1answer
47 views

Prove $Q[x]/(x^2+4)$ is isomorphic to $Q[x]/(x^2+1)$

I've been asked to prove Q[x]/(x^2+4) is isomorphic to Q[x]/(x^2+1); I've looked at lots of similar solutions, but haven't been able to understand this. I know each ring is the quotient ring for their ...
0
votes
1answer
37 views

Characteristic of a product ring?

Let A and B be commutative rings with unity where char(A)=n and Char(B)=m s.t. n,m ∈ ℤ (and n,m ≠ 0). Prove or give counter example: if k ∈ ℤ+ and n,m both divide k, then Char (A x B)| k. Here was ...
0
votes
2answers
37 views

Intersection of two principal ideals is an ideal and lowest common multiple (if it is a PI)

I think the first part of the proof would go like this: any element in $(a) \cap (b)$ can be written as $ar_1 = br_2$, so multiplying by an element $r \in R$ yields $ar_1r\in aR$ or $br_2r \in bR$, so ...
1
vote
2answers
29 views

A commutative unital ring is a field iff its only ideals are $0$ and $R$ [closed]

A commutative ring $R$ with unity is a field if and only if its ideals are $0$ and $R$. How can I prove it?
3
votes
1answer
54 views

How to study for ring theory?

I want to study the theory of rings because it is used when I study representation theory. Here, a ring is not necessarily commutative and doesn't necessarily has unity. I know that there are a few ...
0
votes
2answers
22 views

Definition of ordered ring flawed?

Wikipedia says an ordered ring has these two properties: $a \leq b \rightarrow a+c \leq b+c$ $0 \leq a \land 0 \leq b \rightarrow 0 \leq ab$ Later the article says: $a \leq b \land 0 \leq c ...
-1
votes
0answers
34 views

$\mathbb Z[r_1,r_2,…,r_n] =\mathbb Z[\frac 1m]$ [duplicate]

Question Let $r_1,r_2, ...,r_n \in \mathbb Q $. Then $$\mathbb Z[r_1,r_2,...,r_n] =\mathbb Z[\frac 1m]$$ for some integer $m$. I think m must be the least common multiple of the ...
0
votes
1answer
19 views

Reference on a result about integral closures.

Could you please give a reference or a sketch of a proof for the following proposition? Proposition: The integral closure of a complete local Noetherian domain $R$ is module-finite over $R$ You ...
-1
votes
1answer
26 views

Show that $[E:F] \le n!$ [duplicate]

Let $f(x)$ be a separable irreducible polynomial of degree n with coefficients in a field F. Let E be a splitting field of f(x) over F. Show that $[E:F] \le n!$
1
vote
2answers
37 views

Trying to show that $f$ is not a zero divisor?

Consider the ring $R=\dfrac {k[x,y,z,t]}{(y(xt-yz))}$. Consider the polynomial $f=t(y^3-x^2z)$. Is $f$ a non zero divisors of $R$? How do we check this? I know that if $f$ is a zero divisor then ...
0
votes
1answer
35 views

Is $f$ a non zero divisor?

Consider the ring $R=\dfrac {k[x,y,z,t]}{(y(xt-yz))}$. Consider the polynomial $f=t(yz-xt)$. Is $f$ a non zero divisor of $R$? I think the answer is yes, because if $f$ is zero divisor then ...
1
vote
3answers
54 views

Why does a ring homomorphism not necessarily map unit to unit?

I'm having trouble understanding why in a ring homomorphism, say maps from $R$ to $R'$, doesn't necessarily map the unit $1$ in $R$ to $1'$ in $R'$. If you use the definition that it preserves ...
0
votes
2answers
16 views

How would I work out the Cayley table for $F_3 [x]$ modulo $x^2 +2$ with addition and multiplication.

How would one display the Cayley table for $F_3 [x]/(x^2 +2)$ and show that it is a ring (I have assumed addition and multiplication are associative and that multiplication is distributive over ...
0
votes
1answer
22 views

The existence of a non-split composition series in a indecomposable module

Assume that $R$ is a ring with unit and $M$ is a indecomposable left $R$-module with finite length. That is, $M$ has a composition series. Is it true that there is a composition series ...
3
votes
1answer
68 views

Prove that there exists a normal extension $F/\mathbb{Q}$ with $G(F/\mathbb{Q}) \cong\mathbb{Z}_{5}$.

Prove that there exists a normal extension $F/\mathbb{Q}$ with $G(F/\mathbb{Q}) \cong\mathbb{Z}_{5}$. I tried to solve this problem by thinking about a polynomial which has a splitting field of ...
0
votes
1answer
62 views

How many ideals in a ring R turned into Z/nZ

Say I have a ring R, is there any general way to find out how many ideals it has? I know that if it's a field then there are only 2 ideals, namely (0) and (1), however what if the ring is not a field, ...
2
votes
4answers
102 views

Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb Q$

Prove that $\sqrt[3]{2} +\sqrt{5}$ is algebraic over $\mathbb{Q}$ (by finding a nonzero polynomial $p(x)$ with coefficients in $\mathbb{Q}$ which has $\sqrt[3] 2+\sqrt 5$ as a root). I first tried ...
1
vote
1answer
21 views

If $R/P$ is an integral domain then $P\vartriangleleft R$ is prime. [duplicate]

Let $R$ be a ring and let $P$ be a proper ideal of $R$. If the quotient ring, $R/P$ is an integral domain then $P\vartriangleleft R$ is prime. For $x,y\in R$ we have $(x+P)(y+P)=xy+P\in ...
-1
votes
0answers
14 views

CharK=0 (or p) iff CharF=0 (or p), F is subfield of K [duplicate]

Let $F$ be a subfield of the field $K$. Prove that: 1) $CharK=0 \iff CharF=0$ 2) $CharK=p \iff CharF=p,\ p$ is prime. My thoughts: (a) $1_K \in K$, so $ CharK=ord(1_K) \ | \ |K|$ from Lagrange. If ...
0
votes
1answer
21 views

Fraction rings ideals members

Let $R$ be a ring with fraction ring $R_S$ and ideal $I$. I saw in arguments that when $a/s$ is in $I_S$ they dont say $a$ is in $I$. Instead they say $a/s=b/t$ with $b \in I$. Why? Many thanks.
0
votes
1answer
60 views

Rings where $ab=0$ for all elements

Let $R$ be a ring, not necessarily unital, such that $ab=0$ for all $a,b\in R$. Suppose $R$ only has trivial right ideals. Is it true that $R$ has finite order? Are these rings special?
2
votes
2answers
42 views

Transforming a Polynomial to Show Irreducibility Using Eisenstein's Criterion

I have a particular polynomial $$z^5-5z^4+30z^3-150z^2+465z-725$$ A quick check in mathematica shows that this polynomial is irreducible over the rationals, however, it does not pass the third ...
0
votes
1answer
31 views

Regarding taking powers of prime ideals in a ring

My question is simple to ask: given some prime ideal $P$ in a ring $R$, we can talk about $P^2, P^3$ etc. but can we discuss $P^0$? Is there a convention that says $P^0 = R$, or is there something ...
2
votes
0answers
29 views

What is the ring of integers in $\mathbb Q^c\otimes_K K_\mathfrak p$?

Let $K$ be a number field with ring of integers $\mathcal O_K$ and $\mathfrak p$ a prime of $K$. Let $\mathbb Q^c$ be the algebraic closure of $\mathbb Q$ in $\mathbb C$. If $L$ is a number field ...
0
votes
0answers
16 views

Describe the prime ideals of Ring $R$ in terms of their generators. [duplicate]

Let $R:=\Bbb C[x,y]$ denote the ring of polynomials in the variables $x$ and $y$, with complex coefficients. Describe the prime ideals of $R$ in terms of their generators. Prime ideals are ideals ...
0
votes
3answers
62 views

Show that two rings are not isomorphic [closed]

I don't know how to show (or why) $M_{2\times2}\mathbb{(R)}$ is not isomorphic to $\mathbb{R}[x]/(x^4-1)$ does it have something to do with the order of coset representatives of the quotient group? ...
0
votes
1answer
31 views

Define a new addition ⊕ and multiplication on Z by a⊕b = a + b−1 and ab = a + b−ab.

a+b and ab are the usual integer addition and multiplication. You can assume that this new operation forms a ring, say R is the set of integers with these operations. Then does R have zero-divisors? ...
0
votes
0answers
35 views

Determine the group of units of a subset of $M_n(\mathbb{C})$

Let $R$ be a commutative ring. Let $R=\bigg\{\begin{bmatrix}u & v\\ 0 & u\end{bmatrix}:u,v\in\mathbb{C}\bigg\}$. Determine the group of units $R^{\times}$ of $R$. My try: Let ...
0
votes
0answers
9 views

for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every $r\in R$.

Let be a $R$ ring with a identity. An $R$-module $A$ is injective iff for every left ideal $L$ of $R$ and $R$-module homomorphism $g:L\to A$, there exists $a\in A$ such that $g(r)=ra$ for every ...
1
vote
1answer
43 views

Do all fields have a total cyclic order?

It is well known the finite commutative rings, $Z/nZ$, are not discretely ordered rings. The axiom $\forall x \forall y \forall z((0<z \land x<y) \rightarrow (x*z < y*z))$ is false for the ...
3
votes
2answers
46 views

Multiplicative inverse of $x+f(x)$ in $\Bbb Q[x]/(f(x))$

So I have $f(x) = x^3-2$ and I have to find the multiplicative inverse of $x + f(x)$ in $\mathbb{Q}[x]/(f(x))$. I'm slightly confused as to how to represent $x + (f(x))$ in $\mathbb{Q}[x]/(f(x))$. ...
-3
votes
1answer
35 views

Kernel of a homomorphism: $f(a)=f(x)$? [closed]

Suppose $A$ and $K$ are rings with $f: A \to K$ a homomorphism. Prove that for any $x \in a + \ker(f)$ we have $f(x)=f(a)$. Im not sure how to start this, any help is appreciated!
0
votes
1answer
20 views

Ideal generated by given integers verification.

The question reads: Find the positive generator of the smallest ideal in $\mathbf Z$ containing the following ideals: a. $(4)$ and $(18)$. My answer is $(m)=(4)$. b. $(6)$ and $(35)$. My ...
-1
votes
2answers
32 views

if a is a unit of $A$, it is also a unit of the quotient ring? [closed]

Suppose $A$ is a nontrivial commutative ring with unity and $S$ is an ideal of $A$ s.t. $S\ne\{0_A\}$ and $S\ne A$ . Prove or find a counterexample: if $a\in A$ is a unit in $A$, then $a + S$ is a ...
6
votes
3answers
116 views

Identity of a ring as two different sums of idempotents

Let $R$ be any ring with identity $1_R$. Prove that if there exist idempotents $e_1,..., e_n, e'_1,...,e'_n \in R$ such that $$ 1_R = e_1 +...+ e_n = e'_1+... e'_n$$ then the following conditions ...
2
votes
4answers
53 views

Is $\Bbb Q[x]/(x^2+x)$ isomorphic to $\Bbb Q[x]/(x^2-x)$?

It seems the statement is true, but I have no idea how to prove it I try to let $f=(x^2+x)Q(x)+\bar f=(x^2-2)P(x)+\bar f'$ Then I construct a function $\phi:\Bbb Q[x]/(x^2+x) \to \Bbb ...
-2
votes
1answer
46 views

Prove that $C(X,\mathbb R)$ has no non trivial nilpotent elements. [closed]

Can anybody help me in this question. I have no idea how to proceed. Any HINT will be appreciated.
1
vote
1answer
25 views

Show that the map is bijective.

I have a doubt in part $(3)$ Clearly this map is surjective using part $(b).$ To show injectiveness, let $x\neq y$ To show: $M_x\neq M_y$ Now as we are working in a Hausdroff space so ...
-4
votes
1answer
34 views

Quotient ring $(\mathbb{Z}_4 \times \mathbb{Z}_6)/S$

Consider the ring Z4xZ6 with +6, *6, and +4, *4 in appropriate coordinates and S={(0,0),(2,0),(0,3),(2,3)}. Would the elements of the quotient ring Z4 x Z6 / S be: S+0 (trivial set above), ...
0
votes
0answers
12 views

Linear independence in a module

It is widely known that for any matrix on a commutative field, the following properties are equivalent : 1. Determinant is invertible 2. Matrix has an inverse 3. The only zero linear combinations ...
0
votes
1answer
33 views

Does the group $G$ of $n$th roots of unity form a subring of $\Bbb C$?

Is it true that the group $G$ of $n$th roots of unity is a subring of $\Bbb C$? My initial thought is that this is most definitely not true because the element $0$ is not an $n$th root of unity, and ...
1
vote
0answers
35 views

Polynomial roots in the ring extension

Let $R$ be a ring with identity (not necessarily commutative) and $R[x]$ be a ring of polynomials over $R$. We say that a ring $S$ is an extension of $R$ if there is a subring $\tilde{R}$ in $S$ ...
2
votes
1answer
19 views

Upper Nilradical of a Ring

If we define the upper nilradical of a ring as the sum of all nil ideals of the ring, how could we deduce from just this definition that this is a nil ideal? Thanks!
0
votes
0answers
18 views

What is the difference between ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ and ${\mathbb{Z}[x_1, .., x_n]}/(p)$?

Let $p$ be any prime. What is the difference between ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ and ${\mathbb{Z}[x_1, .., x_n]}/(p)$? ${\mathbb{Z}[x_1, .., x_n]}_{( p )}$ is the localization of ...
1
vote
1answer
18 views

Extending Semiring $\mathbb{N}$ to $\mathbb{Z}$ through exact sequence

I am working on extensions in the form of $$A\hookrightarrow B\twoheadrightarrow C$$ in my thesis and I am just wanting to add as an extra note, IF POSSIBLE, this. We have that that $\mathbb{Z}$ is ...
1
vote
1answer
34 views

Let $M$ be the maximal ideal in $C(X,\mathbb R)$. Prove that there exists $x\in X$ such that $M=M_x$. [duplicate]

I have done part $(a)$ by defining a map from $C(X,\mathbb R) \to \mathbb R $ as $\phi (f)=f(x) $ and got the $M_x$ as kernel of homomorphism and got the answer. But I am unable to solve for ...
10
votes
3answers
87 views

Can we recover a compact smooth manifold from its ring of smooth functions?

It is well-known that if $X$ is a reasonably nice topological space (compact Hausdorff, say) then we can recover $X$ from the ring $C(X)$ of continuous functions $X\to\mathbb R$; see this MSE question ...
2
votes
0answers
23 views

For rings: containment and isomorphism does not imply equality.

So I have been asked to find two commutative unital rings $A$ and $B$ such that $B\subseteq A$ and $A\cong B$ but $A\neq B.$ I give my solution below. I would be very grateful if someone could ...
2
votes
3answers
44 views

$\mathbb{Z} [\sqrt{2}]$ is an integral domain

We know that $(\mathbb{Z} [\sqrt{2}],+,\cdot)$ is an integral domain. Someone can prove it easily if he says that is a subring of $(\mathbb{R} ,+,\cdot)$ . Can we find a different proof, more ...