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
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0answers
54 views

If $\mathfrak{m}\otimes M\rightarrow A\otimes M$ is injective, what else has to be injective?

Let $A$ be a local (not necessarily noetherian) ring with maximal ideal $\mathfrak{m}$ and residue field $k$. Let $M$ be a finitely generated $A$-module such that $\mathfrak{m}\otimes_A M\rightarrow ...
1
vote
1answer
45 views

Find image and kernel of $\varphi: \mathbb{Z}[x] \to \mathbb{C}$ given by $x \mapsto i$

Please help on this, I'm desperate: Consider the homomorphism $\varphi: \mathbb{Z}[x] \to \mathbb{C}$ given by $x \mapsto i$. Find: (a) the image; (b) the kernel; (c) exhibit the bijection of ...
2
votes
3answers
83 views

About the group of units

I'm stuck at this section of the following problem: Let $R$ be a commutative ring and $S$ be a subset of $R$, with $S$ multiplicatively closed. Find the group of units of $S^{-1}R$. My try: ...
6
votes
2answers
89 views
+50

Maximal left ideals $\leftrightarrow$ simple left modules

Suppose $R$ is a ring with unity. This passage in Lang's Algebra discusses the correspondence $$\text{Maximal left ideals of $R$} \leftrightarrow \text{Simple left $R$ modules},$$ where I corresponds ...
1
vote
1answer
40 views

Finding primary decompositions of ideals

I have been given this example of the decomposition of an ideal into primary ideals $$ I =⟨x^2,xy,x^2z^2,yz^2⟩$$ Then the primary decomposition of this ideal is: $$⟨x^2,y⟩∩⟨x,z^2⟩⊆K[x,y,z]$$ This ...
1
vote
1answer
32 views

Show that the variety $V(I(X))=X$

In the ring $R=K[x_1,...,x_n]$, the variety of an ideal is defined as $V(I)=\{(a_1,...,a_n)\in K^n|f(a_1,...,a_n)=0, \space\forall f\in I\}$ The ideal of a variety is defined as $I(V)=\{f\in ...
1
vote
0answers
26 views

ring restriction of linear groups

Suppose that there is a group $G\subseteq\text{GL}(n,\mathbb{C})$ defined to be the group satisfying some equations on $\text{GL}(n,\mathbb{C})$(for example, given a nonsingular matrix $A$, ...
1
vote
2answers
33 views

Show that in a right artinian ring $R$, every prime ideal is a maximal ideal.

Show that in a right artinian ring $R$, every prime ideal is a maximal ideal. **Comments:**I have as a result: For any ring R, are equivalent: (1) R is semisimple; (2) R is semiprime and left ...
0
votes
1answer
30 views

On a question about polynomial ring

Let the ring $ R$ define as the following $R=\{a_1+a_2x^2+a_3x^3+...+a_x^n;a_i\in \mathbb R,\,n\gt 2\}$ and Let the ideal $I$ generated by $<x^2+1,x^3+1>$. Is $I=R$ or not?
3
votes
1answer
54 views

Number of ring homomorphisms form $\mathbb Z[x]$ to $\mathbb Z_{12}$

I have tried : Let $f$ be an homomorphism form $\mathbb Z[x]$ to $\mathbb Z_{12}$. we have to find the possible image of $1$ and $x$. Suppose $f(1) = a$, then $f(1)^2 = f(1) = a^2 = a$, then the ...
0
votes
1answer
36 views

Show the group isomorphism $(\mathbb{Z}/n)^\times \cong (\mathbb{Z}/p_1^{k_1})^\times \times \cdots \times (\mathbb{Z}/p_n^{k_n})^\times$

When $r$ and $s$ are relatively prime we have the ring isomorphism $\mathbb{Z}/rs \cong \mathbb{Z}/r \times \mathbb{Z}/s$ Given a prime factorization of $n$ where $n = p_1^{k_1} \cdots p_n^{k_n}$ ...
1
vote
1answer
33 views

Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field?

Why is it true that any polynomial $f(x)$ with coefficients in a finite field $F_q$ is separable over that field? I'm confused because the polynomial $x^2+1$ in $F_2[x]$ is inseparable ($f(x)$ and ...
4
votes
1answer
63 views

Determining if $\mathbb{Z}[a]$ is a discrete subring of $\mathbb{C}$.

Let $a \in \mathbb{C}$ and consider the ring $\mathbb{Z}[a]$. Is there some nice criterion which will tell me whether $\mathbb{Z}[a]$ is discrete in the sense that there is some $\delta >0$ such ...
0
votes
2answers
42 views

Prove that $v$ and $\overline{v}$ are not associates in the ring $\mathbb{Z}[i].$

Suppose $p$ be an odd prime such that $p \equiv 1 \pmod 4$. In the ring of Gaussian integers $\mathbb{Z}[i]$, $p$ factors as $p = v \cdot \overline{v}$ for a prime $v \in \mathbb{Z}[i].$ ...
0
votes
1answer
54 views

Show that 2Z and 3Z are not isomorphic - question on proof

I need to show that $2\Bbb Z$ and $3\Bbb Z$ are not isomorphic. I found a contradiction as follows: let $p$ be this isomorphism from $2\Bbb Z$ to $3\Bbb Z$. Then $p(4) = p(2*2) = p(2+2)$, so ...
4
votes
1answer
38 views

For a group ring, finding if a subset is an ideal. [closed]

For the ring $R=SG$, the group ring of a finite group G over an integral domain S, and a subset $I=(g-1|g \in G)$, is this subset an ideal? Is it prime? How about maximal?
1
vote
0answers
22 views

Unit or (Left/right zero) divisior [duplicate]

Let R is finite Ring with 1 and $a \in R \setminus \{0\}$. Show that a is Unit or (Left/right) zero divisior It's obvious that we have to use the mapping: $x \rightarrow ax$ and $x \rightarrow xa$ but ...
1
vote
2answers
78 views

Generalization of Chinese Remainder Theorem

Q: With ring $R$, if $I, J \subseteq R$ are ideals such that $I+J=R$, then the map $R/(I \cap J) \to R/I \times R/J$ given by $a + (I \cap J) \mapsto (a+I, a+J)$ is an isomorphism, broadly ...
2
votes
1answer
19 views

Showing a Ring and an Ideal are equal

Q: Let R be a commutative ring with unity. Prove that if A is an ideal of R and A contains a unit, then A=R. This is my attempt at an answer: It suffices to show that all the elements in R are in A. ...
0
votes
2answers
57 views

Proving something it NOT and integral domain

Let $R$ and $S$ be two commutative rings with unity. Prove that $R\times S$ is NOT an integral domain. This is the best I could think of so far, please give me a push in the right direction and ...
4
votes
2answers
24 views

Given a ring with unity and a central idempotent element e, prove some isomorphic relations

Given a ring $R$ with 1 $\neq$ 0, and an element $e$ that is idempotent and central in $R$, I want to prove that $R/Re \cong R(1-e)$, $R/R(1-e)\cong Re$, and subsequently, $R\cong Re\times R(1-e)$. My ...
3
votes
0answers
38 views

Degree of the minimal polynomial of the sum of two integral elements over a UFD

Let $D$ be an integral domain ($D$ is a noetherian UFD, if necessary) and let $a,b$ integral over $D$. Let $f$ be the minimal polynomial of $a$ over $D$ and assume it is of degree $n>1$, and let ...
1
vote
2answers
44 views

Can I say that $R= Rr + I$?

Let $I$ be a maximal left ideal of a ring $R$. I have $y \in radR$ e $r \in R$. I am assuming that $yr \notin I$. Can I say that $R= Rr + I$? Definition: $rad R$ is a intesection of the maximal left ...
1
vote
1answer
26 views

Factorization of polynomial in a ring

I want to show that $x^2-14$ doesn't factorize into $(ax+b)(cx+d)$ in $Z_{2014}$ Since $ac=1$ , $x=c^{-1}(-b)$ or $x=a^{-1}(-d)$ is one of the solution "$f(x)=0$" where $f(x)=x^2-14$ and those are in ...
0
votes
2answers
34 views

Where does the proof of unique factorization fail for $\mathbb Z[\sqrt{-5}]$?

I know that unique factorization does not hold for all rings, such has the much-used example $\mathbb Z[\sqrt{-5}]$. It seems that Euclid's lemma does not hold for these rings, and so on. However, ...
2
votes
1answer
47 views

Show that $(\mathbb{Z}/n)^\times$ cannot be cyclic unless $n$ is either a prime power or twice a prime power.

Q: Show that $(\mathbb{Z}/n)^\times$ cannot be cyclic (i.e., there is no primitive root modulo $n$) unless $n$ is either a prime power or twice a prime power. I'm reading a solution. The first ...
1
vote
0answers
24 views

Reduce $(\mathbb{Z}/n)^\times$ to $(\mathbb{Z}/p^n)^\times$

Please help me understand this problem: Given Problem: Using the ring isomorphism $\mathbb{Z}/rs \cong \mathbb{Z}/r \times \mathbb{Z}/s$, show that we have an isomorphism $(\mathbb{Z}/rs)^\times ...
0
votes
3answers
45 views

Why isn't $\mathbb Z\big[\sqrt d\big]$ a quadratic integer ring when $d \equiv 1 \pmod 4$?

I am trying to understand quadratic integer rings. The following is assuming $d$ is square-free. From my understanding so far: $\mathbb Z\big[\sqrt d\big]$ is an integral domain. $\mathbb ...
0
votes
1answer
58 views

If $d \equiv 1 \pmod 4$, is $\mathbb Q[\sqrt d]$ the field of fractions of $\mathbb Z\left[\frac{1+\sqrt d}{2}\right]$?

If $d \equiv 1 \pmod 4$, is $\mathbb Q[\sqrt d]$ the field of fractions of $\mathbb Z\left[\frac{1+\sqrt d}{2}\right]$? Is $\mathbb Q\left[\frac{1+\sqrt d}{2}\right]$? I am confused about quadratic ...
-3
votes
1answer
58 views

A ring $R$ that is not Artinian but the module $F=R^n$ is Artinian. [closed]

Let $F$ be a free $R$-module of finite rank. I know when $R$ is Artinian ring then $F$ is Artinian $R$-module. Can you give me a ring that is not Artinian but $F$ be Artinian? Thank you very much.
1
vote
1answer
22 views

Is this set necessarily a subring?

I have been doing a lot of work with quadratic fields and I am attempting to generalize the results to abstract fields and rings and I am having trouble showing that a certain set is a subring (I ...
0
votes
0answers
24 views

Identify Ring Isomorphisms

Can someone do a brief sanity check on my work on this: Identify the following rings: (a) $\mathbb{Z}[x]/(x^2-3,2x+4)$ $x^2 = 3 \implies 4x^2 = 12$ $2x=-4 \implies 4x^2 = 16$ $12 \equiv 16 ...
0
votes
1answer
26 views

Let $R$ a left artinian ring. Show that if $a \in R$ is not a right zero divisor, then $a$ is a unit in $R$. [duplicate]

Let $R$ a left artinian ring. Show that if $a \in R$ is not a right zero divisor, then $a$ is a unit in $R$. Comments: I am tryed to do so: See the R-module $_{R}R$ and consider the function $f: ...
2
votes
0answers
48 views

Dimensions of quotient rings of $K[x,y]$

I have tried to solve the following problem and would be very grateful if someone could check my answer. Let $K$ be an algebraically closed field with $\mathrm{char}(K)=0$. I wish to compute ...
2
votes
2answers
62 views

Prove that every prime ideal of $R$ is maximal. [duplicate]

Let $R$ be a commutative ring with identity.For each $a\in R$ there exist $n(>1)\in \mathbb N$ such that $a^n=a$. Prove that every prime ideal of $R$ is maximal. My try Let $I$ be a prime ideal ...
1
vote
1answer
12 views

Difference between an $R$-algebra being finitely generated and finite

So I have the two following definitions: An $R$-algebra $S$ is said to be finite over $R$ if it is finitely generated as an $R$-Module. An $R$-algebra $S$ is said to be finitely generated if $$S ...
0
votes
0answers
24 views

What does it mean for a quotient ring to be finite

This is probably a really simple question, but I couldn't find any equivalent definitions for it online: what do we mean when we say that a Quotient ring is finite? Does it simply mean that that set ...
1
vote
1answer
21 views

Quotient of ideals of the ring of rational numbers with denominator prime to p.

Let $R_p=${ $\frac{m}{n} \in \mathbb{Q} $ | gcd(n,p)=1 } and consider the ideals of $R_p$ : $p^{\nu}R_p$ and $p^{3\nu}R_p$. Then $\frac{p^{\nu}R_p}{p^{3\nu}R_p}$ is a cyclic group of order ...
0
votes
1answer
35 views

Rings, ideals and quotient rings

Suppose $I$ is an ideal of ring $R$, and $J'$ is an ideal of $R/I$. Show there is an ideal $J$ in $R$ so that $J/I=J'$. How do I answer this? What am I required to prove?
0
votes
1answer
27 views

Factorisation of an element into irreducibles - Algebra

I wanted to know if this fact was correct: Suppose we have a ring $R$, and an element $r \in R$. Suppose $r$ has two different factorisations into irreducibles. Then $R$ cannot be a Principal ideal ...
2
votes
1answer
55 views

Under what conditions does a quotient module $A/B\cong C$ imply the direct sum $A\cong B\oplus C$?

Suppose we have some finitely generated $R$-modules $A,\,B,\,C$ such that $A/B\cong C$. Under what conditions is it necessarily true that $A\cong B\oplus C$? Clearly the converse to this is true, but ...
1
vote
0answers
43 views

What is the name of this special subset of module over commutative ring with unity

Let $R$ ba a commutative ring with unity and let $M$ be a $R$-module. For each $m,n\in M$ consider free spaces $\bar{m}$ and $\bar{n}$ generated respectively by $m$ and $n,$ i.e ...
2
votes
1answer
30 views

Intersection and isomorphism of two relatively prime ideals

Let $R$ be a commutative ring. Using the definition that two ideals $I, J \subseteq R$ are relatively prime if $I + J = R$. I want to show that for two relatively prime ideals $I, J \subseteq R$, it ...
1
vote
2answers
26 views

properties of transformations inbetween quotient modules

Let $R$ be a ring, $M$ and $R$-left module and $U, V$ submodules of $M$. I want to show that $$f: M/U \cap V \to M/U \oplus M/V, m + U \cap V \mapsto (m + U, m + V)$$ and $$g: M/U \oplus M/V \to ...
1
vote
0answers
31 views

How to show an ideal is principal

Is there a general procedure to check whether or not a prime ideal of the ring of integers $O_K$ is principal. In my case $K$ is a quadratic field, i.e $\mathbb{Q}(\sqrt {d})$, with $d$ square-free.
1
vote
2answers
48 views

For a maximal left ideal $M$ of $S$, is $f^{-1}(M)$ a maximal left ideal of $R$ when $f$ is surjective?

Let $f:R \longrightarrow S$ a surjective ring homomorphism. Is the inverse image $f^{-1}(M)$ a maximal left ideal of $R$ for any maximal left ideal $M$ of $S$? Comments: I tied something ...
0
votes
1answer
29 views

existence of module homomorphisms, so that a diagram becomes commutative

Let $R$ be a ring, $f: R^k \to R^m$ aswell as $g: R^l \to R^n$ module homomorphisms and $M = coker(f)$, $N = coker(g)$. Let $p_M: R^m \to M$ and $p_N: R^n \to N$ be the natural projections. I now ...
0
votes
0answers
27 views

Writing a group as a product of cyclic groups

How do I go about expressing a group as a product of cyclic groups? For example, express: $$O^*_K = \{\pm (24 + 5\sqrt{23})^r : r\in \mathbb{Z} \}$$ as a product of cyclic groups ($O^*_K$ is the ...
1
vote
1answer
46 views

Some weaker axiom than “no nontrivial zero divisors.”

I would like to know if there a standard term for or well-known applications of the following axiom for rings or semigroups with zero (which is weaker than the "no nontrivial zero divisors" axiom): ...
1
vote
1answer
30 views

Characteristic of Quotient Ring

Consider a ring $R$ with characteristic $n \gt 0$. Let $I$ be an ideal of $R$ with characteristic $m$. I have proved that $m$ divides $n$. Now I am interested about the characteristic of $R/I$. One of ...