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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5
votes
1answer
84 views

What is the kernel of $K[x^2,x^3][T] \to K[x]$, defined by: $T \mapsto x$?

Consider $K[x^2,x^3] \subset K[x]$, where $x$ is an indeterminate over a (zero characteristic) field $K$. Clearly, $x$ vanishes the following polynomials $\in K[x^2,x^3][T]$: $f(T)=x^2T-x^3$, ...
1
vote
0answers
14 views

converging power series over $p$-adic integers is a UFD

Denote by $|\cdot |$ the $p$-adic norm, and let $$\mathbb Z_p \{z\}=\left\{\sum_{n=0}^\infty a_nz^n;\ a_n\in \mathbb Z_p ,\ |a_n|\underset {n\rightarrow \infty} {\longrightarrow 0} \right\}$$ the ...
1
vote
1answer
79 views

In what structures does $ (-1)^2 = 1$?

Does $ (-1)^2 = 1$ anywhere where you have associativity and an inverse element? Thanks!
9
votes
2answers
84 views

For which $d \in \mathbb{Z}$ is $\mathbb{Z}[\sqrt{d}]$ a unique factorization domain?

Is there a general criterion which tells me whether $\mathbb{Z}[\sqrt{d}]$, $d \in \mathbb{Z}$ is a unique factorization domain? $\mathbb{Z}[\sqrt{-5}]$ is a frequent example for non-unique ...
1
vote
0answers
50 views

searching about an ismorphism

I'm looking for an isomorphism : $$H: \overbrace{\mathbb{F}_q^r\oplus\cdots\oplus \mathbb{F}_q^r}^{l\text{ times}}\longrightarrow \frac{\mathbb{F}_q[X_1,\ldots,X_l]}{(X_1^r-1)\cdots(X_l^r-1)}$$
3
votes
1answer
39 views

Looking for a terminology in ring theory (“ideal” which is not necessarily closed under addition )

I am wondering if there is a name for the subsets $S$ of a commutative ring $R$ such that for every $r\in R$ and every $s\in S$ we have $rs\in S$. Thus $S$ is ...
2
votes
0answers
34 views

Polynomial-closed properties of rings

If $R$ is a ring with certain property, sometimes when we pass to the polynomial ring in one variable, the ring $R[x]$ still has the same property. For instance, it's a theorem that if $R$ is a UFD ...
2
votes
1answer
69 views

When an Intersection of Prime Ideals is a Prime Ideal

Let $R$ be an arbitrary ring, $\{P_1,....,P_n\}$ be a set of prime ideals. Verify that $P_1 \cap ... \cap P_n$ is prime if and only if there exists $1 \leq i \leq n$ such that $P_i$ is contained in ...
22
votes
5answers
987 views

Why is the commutator defined differently for groups and rings?

The commutator of two elements in a group is defined as $[g, h] = g^{−1}h^{−1}gh.$ In a ring, the commutator of two elements is $[a, b] = ab - ba.$ I'm asking because a ring is a (abelian) group ...
3
votes
1answer
39 views

Is a = 0 a valid counterexample to this statement?

This is an exercise in a text I am reading for a ring theory course. Suppose the ring R contains element a such that 1) a is idempotent and 2) a is not a zero divisor of R. Deduce that a serves as a ...
4
votes
1answer
67 views

Under what conditions does $M \oplus A \cong M \oplus B$ imply $A \cong B$?

This question is fairly general (I'm actually interested in a more specific setting, which I'll mention later), and I've found similar questions/answers on here but they don't seem to answer the ...
1
vote
1answer
43 views

What is the difference between submodules of $A/\mathfrak a$ as an $A$-module or as an $A/\mathfrak a$-module?

If $\mathfrak a$ is an ideal of unital commutative ring $A$, Then we can see to $A/\mathfrak a$ as an $A$ module or as an $A/\mathfrak a$ module. If $A=\mathbb Z$ there is no structure difference ...
1
vote
1answer
46 views

Show that every short exact sequence $0\rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0$ (with $M''$ free) is split exact

Note: $M,M',M''$ are modules. In order to show that it's split exact, I understand that I need to show that $\beta:M\rightarrow M''$ has a left inverse, and similarly that $\alpha:M'\rightarrow M$ ...
7
votes
2answers
145 views

Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s

Motivated by this question, I want to find a complete set of relations for the ring of integer-valued polynomials, where the generators are the polynomials $\binom{x}{n}$ for $n\in \mathbb{N}$. The ...
8
votes
1answer
68 views

$A$ regular, $k'/k$ transcendental. How to prove that $A \otimes_k k'$ is regular?

Let $k$ be a field and $k'$ a purely transcendental extension of $k$. Let now $A$ be an integral finitely generated $k$-algebra. How to prove that if $A$ is regular then $A \otimes_k k'$ is also ...
4
votes
3answers
446 views

Example of a ring with infinitely many zero divisors and finitely many invertible elements

I am preparing to my abstract algebra exam and I try to find an example of a ring with infinitely many zero divisors and finitely many invertible elements (rather simple if possible). Does it even ...
0
votes
0answers
20 views

recover (pontrjagin) ring structure from the localization (w.r.t. $\pi_0$)

Let $R$ be a ring and $S$ a given multiplicative subset of $R$. Suppose we know the multiplication structure of $S$. If we know the ring structure of $R[S^{-1}]$, the localization of $R$ with respect ...
1
vote
1answer
28 views

Example of a Non-Graded Ideal in a Graded Ring

A ring $S$ is said to be graded if there are additive subgroups $S_0, S_1, S_2, \ldots$ such that $S=\bigoplus_{k\geq 0}S_k$ and $S_iS_j\subseteq S_{i+j}$ for all $i$ and $j$. An ideal $I$ in a ...
3
votes
2answers
72 views

The subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$ is not finitely generated over $k$ [duplicate]

Let $R$ be the subring of $k[x, y]$ generated by $\{x y^{i}: i\geq 0\}$. Can someone explain why $R$ is not finitely generated as a ring over $k$ (i.e. finitely generated as a $k$-algebra)? By ...
2
votes
0answers
19 views

Mixed vs Equal Characteristic Local Rings

This is more of a vague, intuitive type of question, so perhaps there isn't anything too concrete anyone can offer. I am trying to get a sense of precisely why working with local rings of equal ...
2
votes
1answer
28 views

Example of $Q((x))$ that doesnt match field of fractions of ring $F[[x]]$

Let $F$ be a commutative ring without zero divisors and $Q$ -its field of fractions. Let $Q(x)$ be also field of fractions of ring $F[x]$. How can field $Q((x))$ not match field of fractions of ring ...
3
votes
1answer
106 views

What motivates the definition of a ring in abstract algebra? [on hold]

I've read a lot of questions about rings but I still don't know why they are useful/ I'm sure they are, but I don't know why. Are their properties somehow used in proofs or as foundations for ...
2
votes
2answers
33 views

There is no nontrivial ring homomorphism between two commutative rings with unity and characteristic of distinct primes

The following is an old exam question and the question is: Show that there is no nontrivial ring homomorphism between two commutative rings with identity if their characteristics are distinct primes. ...
1
vote
1answer
35 views

Is there a way to generate groups, rings, fields, etc.? [on hold]

There are ways to generate a list of numbers $a_1, a_2,...a,n$ such that no $a_i,a_j$ share any factors, mainly by letting $a_{n+1}=a_n^2-a_n+1$ with $a_0>1$, my question is: is there a way to ...
1
vote
1answer
41 views

Prime ideals in $R[x]$, $R$ a PID

Let $R$ be a PID. Show that if $r \in R$ and $$p = (r, \underline{f}(x), \underline{g}(x))$$ is prime, where $\underline{f}(x), \underline{g}(x) \in R[x]$ are nonconstant irreducible polynomials, ...
5
votes
2answers
78 views

Prime ideal $P$ of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z}=\{0\}$ is principal

The problem stated more precisely is this: Let $P$ be a prime ideal of $\mathbb{Z}[x]$ such that $P \cap \mathbb{Z} =\{0\}$. Show that $P$ is a principal ideal. I think there is a problem with my ...
2
votes
1answer
47 views

Example of a ring of finite global dimension with flat qu0tient

I've been thinking about this for quite a while but I cannot seem to find an example of If $k$ is a commutative ring of finite global dimension and I'm looking for a strictly not-commutative ...
1
vote
2answers
40 views

The ideal generated by the set $I \cup \{a\}$.

Let $I$ be an ideal of $R$ a commutative ring with identity. For some $a \in R$, prove that the ideal generated by $I \cup \{a\}$, denoted $(I,a) = \{ i + ra \mid i \in I \text{ and } r \in R\}$. My ...
1
vote
1answer
113 views

Projective dimension of all principal ideals is finite. Is R an integral domain?

$R$ is a noetherian ring in which projective dimension of all principal ideals is finite. Is $R$ an integral domain? What condition can be added on it to be a regular ring? thanks for any help. ...
3
votes
1answer
93 views

If $R$ is a noetherian local ring, then every 2-generated ideal has finite projective dimension iff $R$ is a UFD

This question is about zcn's comment on the answer to this question. It's a good point. So I ask it for use of everybody: if $R$ is a noetherian local ring, then every 2-generated ideal has ...
3
votes
1answer
30 views

The quotient of a direct sum of rings

Is the quotient of a direct sum of rings isomorphic to the direct sum of the quotients? $$ (R_1 \oplus R_2 ) / \langle (x_1, x_2)\rangle = (R_1 / \langle x_1\rangle ) \oplus (R_2 /\langle x_2\rangle) ...
2
votes
1answer
249 views

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PIR.

Let $f:R\longrightarrow S$ be a surjective ring homomorphism. If $R$ is PID, then $S$ is PIR. I think I have proved this: Let $J$ be an ideal of $S$. Then $f^{-1}(J)=(a)$ is a principal ideal of ...
3
votes
4answers
50 views

If $R$ is a ring with identity and $a$ is a unit, prove that the equation $ax=b$ has a unique solution in $R$.

So, this was my initial proof: Assume $R$ is a ring, and $a,b\in R$ Let $x_1$ and $x_2$ be solutions of $ax=b$ Hence, $ax_1=b=ax_2 \Rightarrow ax_1-ax_2=0_R \Rightarrow a(x_1-x_2)=0_R$ Thus, we ...
3
votes
2answers
50 views

Can one decompose any ring into a (possibly infinite) product of indecomposable rings?

Let $R$ be a ring. Do there exist indecomposable rings $R_i$, $i\in I$, such that $R={\prod}_{i\in I} R_i$?
1
vote
1answer
83 views

$f(x) \mid g(x) \iff g(x) \in \langle f(x) \rangle$. Isn't this trivial?

Let $F$ be a field and $f(x), g(x) \in F[x]$. Show that $f(x)$ divides $g(x)$ if and only if $g(x) \in \langle f(x) \rangle$. This seems... almost trivial to me (which is usually a sign that I'm ...
3
votes
1answer
72 views

global dimension of rings and projective (flat) dimension of modules

Let $R$ be ring such that every left $R$-module has finite projective dimension ( resp. finite injective dimension). Is the left global dimension of $R$ finite? Similarly, Let $R$ be ring such that ...
0
votes
1answer
38 views

Show that a Set S is a subring of $R \times R$

Question: Prove that$$S=\{ (r,r) | r \in R\}$$ is a subring of $R \times R$. Attempt: Proof: Let $(a,b)$ and $(c,d)$ $\in R$. As $(a,b) \cdot (c,d) = (ac,bd) \in S$. $(a-c, b-d) \in S$. I show ...
0
votes
1answer
19 views

What is a relation (finitely related module)?

https://en.wikipedia.org/wiki/Finitely-generated_module#Finitely_presented.2C_finitely_related.2C_and_coherent_modules I've understood the first part of the definition. Then, "M is isomorphic to ...
3
votes
3answers
63 views

Silly question about group rings

Let $R$ be a ring and $G$ be a finite group, and $RG$ be the group ring. What does it mean to say that $|G|$ is invertible in $R$? Since $|G| \in \mathbb{N}$, $|G|$ is not an element of $R$, does it ...
3
votes
1answer
57 views

Isomorphic or equal?

Let $\sim_n$ be the usual equivalence relation of congruence modulo $n$ in $\mathbb{Z}$, i.e., for $a,b\in\mathbb{Z}$, $a\sim_nb\Leftrightarrow a-b=k\cdot n$ for some $k\in\mathbb{Z}$. For $n=0$ the ...
2
votes
2answers
52 views

Structure of the unit group $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$

I know that $\mathbb{Z}[i]/8\mathbb{Z}[i]=\{a+ib \mid a,b\in\mathbb{Z}_8\}$. But I'm not able to comprehend what $(\mathbb{Z}[i]/8\mathbb{Z}[i])^\times$ is. Can someone please help me get its ...
2
votes
4answers
209 views

Global dimension of free algebra.

Is there any easy way to see the global dimension of a free algebra $$ A=k\langle x_{1},\dots,x_{n} \rangle $$ is 1?
0
votes
1answer
13 views

I is equal to the preimage of its image.

Lemma. Let $f$ be a homomorphism from the ring $R$ onto the ring $R'$. If $I$ is any ideal of $R$ such that $\ker(f)$ is a subset of $I$, then $I = f^{-1}(f(I))$. I am trying to understand this ...
0
votes
4answers
39 views

If $I$ and $J$ are distinct ideals in ring $R$ and $f:R \to R'$ is a homomorphism then is $f(I) = f(J)$?

The text book I am reading says that if $I$ is a subset of $J$ and $J$ is a subset of $I + \ker (f)$ then $f(I) = f(J)$. The argument goes: $f(I)$ is a subset of $f(J)$ is a subset of $f(I + \ker (f)) ...
2
votes
1answer
21 views

is $\mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \{1 \}$ divisible subgroup of $ \mathbb{Q} ^{|\mathbb{R}|} ‎\times‎ \mathbb{Z}_2$?

According to Unit Groups of Classical Rings by Karpilovsky, p.107 we know that: If $F$ is a real-closed field, then $F^*‎\simeq‎ \mathbb{Q} ^{|F|} ‎\times‎ \mathbb{Z}_2$. Now, we know that ...
0
votes
0answers
35 views

Prime ideal in indecomposable commutative ring [on hold]

Let $R$ be a commutative indecomposable ring with Jacobson radical $J$. When can we find a prime ideal contained in $J$?
3
votes
2answers
76 views

Semilocal commutative ring with two or three maximal ideals

Is there any equivalence condition for a commutative ring to have exactly two or three maximal ideals?
0
votes
0answers
27 views

Proof that the kernel is a normal subgroup of the domain: repeated line

On proofwiki (https://proofwiki.org/wiki/Kernel_is_Normal_Subgroup_of_Domain), the lines corresponding to 'definition of identity' and 'definition of kernel' are identical. Why do we need the second ...
3
votes
1answer
41 views

A ring isomorphic to its square [duplicate]

Is there an example of a ring $A$ (with unity) which is isomorphic as unital rings to $A\times A$? Any such ring can't have invariant basis number so in particular can't be commutative.
1
vote
1answer
55 views

Prove that $f$ is a nonzerodivisor on $R[x_1,\dots,x_r]/IR[x_1,\dots,x_r]$ for every ideal $I$ in $R$

Let $R$ be a Noetherian commutative ring with unity, and $S=R[x_1,\dots,x_r]$. Let $f\in S$ be a nonzerodivisor of $S$. Suppose that the ideal generated by the coefficients of $f$ is $R$. How to ...