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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1
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2answers
62 views

gcd of $x$ and $2$ in $Z[x]$

In $Z[x]$, $x$ and $2$ has gcd $1$. But they cannot be expressed as the linear combination of two polynomials. Then assuming that $1=2.f(x)+x.g(x)$ we are supposed to arrive at a ...
0
votes
1answer
23 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
25 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 ...
2
votes
3answers
1k views

If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal

If A and B are ideals of a ring, show that A + B = $\{a+b|a \in A, b \in B\}$ is an ideal I have the ideal test but no clue as to what to do with it: $a-b \in A$ whenever $a,b \in A$ $ra$ and $ar$ ...
1
vote
0answers
158 views

Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R. [duplicate]

Let $I, J$ be ideals of a ring $R$. Show that $I + J = \{a + b\vert a \in I, b \in J\}$ is an ideal of R. Because $I,J$ are ideals of $R$, so $I,J$ both have $0$, thus $0+0=0\in I+J$. This shows ...
0
votes
0answers
7 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
votes
1answer
93 views

If $x,y$ are nilpotent and commute, $x+y $ is nilpotent. [closed]

Let $A$ a ring. Supose that $x,y \in A$ are nilpotents elements and that $xy=yx$. Prove that $x+y$ is nilpotent.
14
votes
2answers
840 views

In which algebraic setting can I state (and prove) the binomial theorem?

In a book on algebra I'm currently working with a proof that uses the binomial theorem for $(x+y)^m$ where $x,y$ are elements of some arbitrary field $k$. This looks strange to me, so I did some ...
0
votes
1answer
18 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 ...
-3
votes
1answer
29 views

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

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!
1
vote
1answer
23 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 ...
1
vote
1answer
22 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 $\mathbb{Q}[x]/(f(x))$. ...
0
votes
1answer
17 views

prove the unity cannot map to the zero in a ring homomorphism? [on hold]

If $A$ and $K$ are nontrivial rings and $f: A \to K$ is an onto ring homomorphism then $f(1_A)\neq 0_K$. My idea was to try to use the kernel somehow, but I'm not sure how to show this.
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votes
2answers
30 views

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

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 ...
2
votes
0answers
15 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 ...
0
votes
1answer
39 views

The ideal is maximal - Show the inclusion

I want to show that the ideal $(x,y)$ is maximal in $F[x,y]$ and that it holds that $(x,y)^2\subseteq (x^2, xy, y^2)\subset (x^2, y)\subset (x,y)$. Knowing that the principal ideal $(x)$ is ...
2
votes
4answers
50 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 ...
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1answer
39 views

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

Can anybody help me in this question. I have no idea how to proceed. Any HINT will be appreciated.
3
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0answers
19 views
+50

On those integers $n>1$ such that any commutative ring with identity having exactly $n$ ideals is a PIR

Convention : All rings are commutative with unity unless stated otherwise. By ideals we will mean to include $\{0\}$ and $R$ also. Let us call an integer $n>1$ a "principal number" if any ring ...
1
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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 ...
6
votes
1answer
78 views
+50

$R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ Noetherian?

Let $R$ be a commutative ring with unity such that every surjective ring homomorphism $f:R\to R$ is injective , then is $R$ a Noetherian ring ?
3
votes
1answer
95 views

Proving Extended Eisenstein Criterion

I need to prove this extended Eisenstein criterion. Let $f(x)=a_n x^n + \cdots + a_mx^m +\cdots+ a_1x + a_0\in\mathbb Z[x]$ be given. If for $p$ prime, $p$ does not divide $a_m$, $p$ divides $a_i$ ...
0
votes
2answers
24 views

On those integers $n>1$ such that there exists a commutative ring with identity with exactly $n$ ideals

Let $n>1$ be an integer; we call $n$ a "ring number" if there exists a commutative ring $R$, with identity, having exactly $n$ ideals (including $\{0\}$ and $R$); now since for every $n>1$, ...
1
vote
1answer
27 views

Modules over rings which are NOT a PID, or NOT a UFD [on hold]

I am interested in studying the properties of modules over rings which are not Principal Ideal Domains or are not Unique Factorization Domains, but I am finding it very difficult to find any material ...
-4
votes
0answers
21 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), ...
1
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1answer
15 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 ...
0
votes
1answer
70 views

Assume that $I/J$ is a prime ideal of $R/J$, is $I$ a prime ideal of $R$? [closed]

$R$ is a commutative ring. $I$ and $J$ are ideals of $R$ with $J\subseteq I$. Assume that in the quotient ring $R/J$, $I/J$ is a prime ideal. Is the ideal $I$ a prime ideal in $R$?
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0answers
34 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 $S$ ...
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0answers
11 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
25 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 ...
9
votes
3answers
72 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 ...
-4
votes
2answers
47 views

Set theory with abstract algebra [on hold]

Let R be an integral domain and let $x,y \in R$ nonzero. Prove that $xR = yR$ if and only if $y = xu$ for some unit $u$. Been having a lot of trouble with this. I'm not very good at this abstract ...
2
votes
1answer
15 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
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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
32 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 ...
1
vote
4answers
32 views

Isomorphism of a ring of matrices

Is it possible for a ring of matrices to isomorphic a ring of numbers? Suppose $$R = \begin{pmatrix} a & b \\ -3b & a \\ \end{pmatrix} a,b \in \mathbb Z $$ Can $R$ ...
0
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1answer
28 views

Can we find ( characterize ) all non-zero commutative Artinian rings $R$ for which $-1 , 1$ are the only units of $R$ ?

Can we find ( characterize ) all non-zero commutative Artinian rings $R$ for which $-1 , 1$ are the only units of $R$ ?
2
votes
0answers
22 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
41 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 ...
9
votes
5answers
4k views

Prove that the Gaussian Integer's ring is a Euclidean domain

I'm having some trouble proving that the Gaussian Integer's ring ($\mathbb{Z}[ i ]$) is an Euclidean domain. Here is what i've got so far. To be a Euclidean domain means that there is a defined ...
2
votes
2answers
45 views

Are Z and Z* (defined below) isomorphic as rings?

Define $\mathbb{Z}^*$ to be the set of integers but with the following operations: $a \circ b = a + b - 1$ and $a * b = a + b - ab$ where $a+b$ and $ab$ are the usual integer addition and ...
1
vote
1answer
40 views

Integral extension and s.o.p.

Let $R\subset S$ be an integral extension. Is a system of parameters of $R$ a system of parameters of $S$ and conversely? I think so, since there is good behavior in dimensions. Many thanks.
1
vote
1answer
62 views

Do we have to show that $f(x)\in R$?

Let $R$ be a commutative ring with unity. I want to show that if $g(x)=c_nx^n+\dots+c_0\in R[x]$ is a zero divisor of $R[x]$ then there exists $d\in R \setminus \{0\}$ such that $dc_n=dc_{n-1}=\dots ...
1
vote
1answer
23 views

Sum of nil right ideals as an ideal

I have two questions: 1) If $S$ is the sum of all right nil ideals of a ring $R$ (with unity), is it true that $S$ is a two-sided ideal? It is clear for me that $S$ is a right ideal (and it is nil if ...
0
votes
0answers
20 views

Is there any proof of the criterion of determining maximal ideal in a commutative ring with unity by Third Isomorphism Theorem?

Theorem. Let $R$ be a commutative ring with unity $1$ and $M$ is an ideal of $R$. Show that $M$ is maximal iff $\dfrac{R}{M}$ is a field. In the proof of this theorem the methods so far I have ...
0
votes
0answers
18 views

$\mathbb{Z}[\frac{1-\sqrt{-19}}{2}]$ - principal ring, but not an euclidean ring [duplicate]

I am stucked on the problem. Is there someone who could tell me why $\mathbb{Z}[\frac{1-\sqrt{-19}}{2}]$ is a principal ring, but it is not an euclidean ring?
-1
votes
1answer
31 views

Factoring polynomials in $\Bbb Z_n$

a). Factor $f(x) = x^3 + 4x^2 + 5x + 2$ completely over $\Bbb Z_7$. b). Give two different factorizations of $x^2 + x + 8$ in $\Bbb Z_{10}[x]$. I have found the zeros of both of these but I am ...
0
votes
1answer
56 views

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$

Find all the zeros of $f(x) = x^3 + 3x + 5$ in $\Bbb Z_7$. I've tried factoring this into multiple forms but I can't seem to find an easy way to find the $x'$s for $x^3 + 3x + 5 = 0$. Any hints or ...
0
votes
1answer
21 views

Two questions regarding polynomial rings.

Give an example of a natural number $n > 1$ and a polynomial $f(x) ∈ \Bbb Z_n[x]$ of degree $> 0$ that is a unit in $\Bbb Z_n[x]$. For this is set $n=2$. So then $f(x) = x \in \Bbb Z_2[x] $. ...
0
votes
2answers
46 views

General questions about Polynomial Rings [on hold]

I'm learning about polynomial rings in my class. My instructor and book are both spectacularly unhelpful and didn't even bother to define most of the terms in my homework. So I have some general ...