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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7
votes
3answers
711 views

Help with proof that $\mathbb Z[i]/\langle 1 - i \rangle$ is a field.

I have been having a lot of trouble teaching myself rings, so much so that even "simple" proofs are really difficult for me. I think I am finally starting to get it, but just to be sure could some one ...
2
votes
1answer
57 views

Morita contexts and Noetherianity/affineness

Let $(R\,,\, S\,,\, _RM_S\,,\, _SN_R\,,\, f\,,\, g)$ be a Morita context with $NM=S$ and $R$ right Noetherian. Show that $S$ is right Noetherian as well. If we further assume $R$ is an affine ...
1
vote
0answers
81 views

Finitely generated ideal question.

Suppose $R$ is a ring, $I \subset R$ is an ideal, and $I = \langle S \rangle$ is finitely generated where $S \subset R$. Show that if $I$ and $J$ are finitely generated ideals of $R$, then so are $I ...
8
votes
1answer
106 views

If a tensor product is free, what can we say about the tensor factors?

Here is what I'd like to prove: Let $R$ be a commutative, noetherian ring, and let $M$ and $N$ be finitely generated $R$-modules. Suppose $M\otimes_RN\cong R$. Does it follow that $M\cong N\cong ...
1
vote
1answer
530 views

Show that every ideal of the matrix ring $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$ [duplicate]

Suppose $R$ is a commutative ring. Show that every ideal of $M_n(R)$ is of the form $M_n(I)$ where $I$ is an ideal of $R$. I have spent 30 minutes on this question and I still got nowhere. Can ...
1
vote
1answer
123 views

Artinian ring with zero finitistic dimension

Let $R$ be a left artinian ring with identity. Suppose $R$ contains copies of all its simple right $R$-modules. Is it true that every left $R$-module of finite projective dimension is projective (so ...
2
votes
4answers
131 views

Show that if $c_1 + c_2\sqrt{5}$ divides $n$ in ${\bf{O}}[\sqrt{5}]$, then so does $c_1 - c_2\sqrt{5}$

I have a ring: $${\bf{O}}[\sqrt{5}] = \{c_1 + c_2\sqrt{5}: (c_1 \in \mathbb{Z} \wedge c_2 \in \mathbb{Z}) \lor (c_1 + \frac{1}{2} \in \mathbb{Z} \wedge c_2 + \frac{1}{2} \in \mathbb{Z}) \}.$$ I ...
2
votes
1answer
85 views

Whether a domain is Dedekind or not

We know from algebraic number theory that if $d$ is a square-free integer, $d\neq 0,1$ and $d$ is congruent to $2,3$ modulo $4$ then $\mathbb{Z}[\sqrt{d}]$ is the ring of integers of the quadratic ...
2
votes
2answers
79 views

Show that the ideals of $\mathbb Z$ are principal.

Exercise: Show that every ideal $I$ of $\mathbb{Z}$ is principal. Attempt: Since $I$ is principal, it can be generated by one element. Also, my tutor said that if $I \subset \mathbb{Z}$ is an ideal ...
1
vote
4answers
439 views

Help with proof that $I = \langle 2 + 2i \rangle$ is not a prime ideal of $Z[i]$

(Note: $Z[i] = \{a + bi\ |\ a,b\in Z \}$) This is what I have so far. Proof: If $I$ is a prime ideal of $Z[i]$ then $Z[i]/I$ must also be an integral domain. Now (I think this next step is right, ...
2
votes
2answers
98 views

Proof: let $A$ a ring, then $(-a) \cdot (-b) = a \cdot b $ $\forall a,b \in A$

I must prove this property: Property: let $A$ a be ring, then $(-a) \cdot (-b) = a \cdot b $, $\forall a,b \in A$. Proof: let $a \in A$ and $b \in A$, by hypothesis $A$ is a ring then $a \cdot 0=0$ ...
0
votes
1answer
63 views

Ring of fractions in $\mathbb{Z}/35\mathbb{Z}$

How can I determine $S^{-1}(\mathbb{Z}/35\mathbb{Z})$, where $S$ consists of of all elements of $\mathbb{Z}/35\mathbb{Z}$ except $0,5,10,15,20,25,$ and $30$?
2
votes
2answers
770 views

What is “prime factorisation” of polynomials?

I have the following question: Find the prime factorization in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreducibility in $\mathbb{Z}[x]$, of ...
1
vote
3answers
177 views

Ring homomorphism question.

If $R$ is a ring, show that there is exactly one ring homomorphism $\phi: \mathbb{Z} \to R$. I can't grasp the idea that there can be only one ring homomorphism. Aren't there many (at least more than ...
3
votes
6answers
382 views

Show that $(2+i)$ is a prime ideal

Consider the set Gaussian integer $\mathbb{Z}[i]$. Show that $(2+i)$ is a prime ideal. I try to come out with a quotient ring such that the set Gaussian integers over the ideal $(2+i)$ is either ...
3
votes
1answer
791 views

Prove that $R$ is a commutative ring if $x^3=x$ [duplicate]

Let $R$ be a ring satisfying: $\forall x\in R, \; x^3=x$. Prove that $R$ is a commutative ring.
2
votes
2answers
56 views

Questions regarding Rings.

I barely passed abstract algebra when I was in college, and 3 years later I bought a book and studied on my own. And currently I am having trouble with Rings with certain conditions. Let $\mathbb ...
2
votes
1answer
67 views

About injectivity of induced homomorphisms on quotient rings

Let $A, B$ be commutative rings with identity, let $f: A \rightarrow B$ be a ring homomorphism (with $f(1) = 1$), let $\mathfrak{a}$ be an ideal of $A$, $\mathfrak{b}$ an ideal of $B$ such that ...
3
votes
1answer
97 views

A torsor equivalent for a ring

Reading John Baez's essay on torsors, I was quite intrigued with the last section which states: "Finally, one more remark for people who want to go further. Near the beginning of this essay, I ...
1
vote
1answer
49 views

Relations between change of ring and projectivity/injectivity

1) If $ P $ is $A$-projective and $ f : A \to B $ is a ring homomorphism then $ B \otimes P $ is $B$-projective ? 2) If $M$ is $A$-injective and $ f : A \to B $ is a ring homomorphism then $ ...
0
votes
2answers
125 views

Hom functors and exactness

Is it true that the sequence $ M \to N \to P $ of $A$-modules is exact if the induced sequence $$\mathrm{Hom}_{A}(F, M) \to \mathrm{Hom}_{A}(F,N) \to \mathrm{Hom}_{A}(F,P) $$ and/or the sequence ...
1
vote
1answer
28 views

simply polar elements in a ring

An element $a$ in a ring $A$ with identity is said to be simply polar if there is $b$ for which $a=aba$, with $ab=ba$. If in addition $b=bab$ then such an element $b$ is unique. The question is ...
0
votes
2answers
41 views

cardinality of elements in a “semiring minus multiplicative identity”

In a theory that has all axioms of semiring except multiplicative identity axiom, will there be a model of the theory that has infinite elements? The model must violate multiplicative identity axiom.
2
votes
2answers
77 views

Name for a semiring minus multiplicative identity requirement

Is there a name for a theory that has all axioms of a semiring but an axiom that mandates multiplicative identity?
0
votes
2answers
65 views

Find a zero divisor in $Z_7 [x]/I.$

Let $f (x) ∈ Z_7 [x]$ be the polynomial $x^2 + [3]x + [3]$ and let $I$ denote the principal ideal generated by $f (x).$ Find a zero divisor in $Z_7 [x]/I.$
1
vote
1answer
47 views

Is $\{x\in R\mid A \cap Rx=\emptyset\text{ and }A \cap xR=\emptyset\}$ infinite in a ring?

Assume $R$ is a ring and $A\subseteq R$ contains $0$. Let $$B=\{x\in R\mid A \cap Rx=\emptyset\text{ and }A \cap xR=\emptyset\}$$ Can $B$ be nonempty? If $B$ is nonempty, is it infinite?
0
votes
2answers
37 views

In $Z_5 [x]$, the monic GCD of the polynomials $(x+[4])(x+[3])$ and $([3]x + [2])(x + [3])$ is $(x + [3])$

True or False In $Z_5 [x]$, the monic GCD of the polynomials $(x+[4])(x+[3])$ and $([3]x + [2])(x + [3])$ is $(x + [3])$. my solution : $([3]x+[2])$ is $[3](x+[4])$ therefore gcd is ...
110
votes
0answers
5k views

A short proof for $\dim(R[T])=\dim(R)+1$?

If $R$ is a commutative ring, it is easy to prove $\dim(R[T]) \geq \dim(R)+1$. For noetherian $R$, we have equality. Every proof I'm aware of uses quite a bit of commutative algebra and non-trivial ...
3
votes
1answer
54 views

Find $(1-ba)^{-1}$ when $c=(1-ab)^{-1} $ in ring $R$.

For $R$ is a ring has identity element. $a,b\in R$ and $c=(1-ab)^{-1}$ . Find $(1-ba)^{-1}$.
0
votes
0answers
52 views

number of Ring homomorphism [duplicate]

The number of non-trivial ring homomorphism from $\mathbb Z _{12}$ to $\mathbb Z _{28}$. Is there any general formula for ring homomorphism between $\mathbb Z _{m}$ to $\mathbb Z _{n}$, like we have ...
4
votes
1answer
64 views

Reference request: Morita contexts

During an independent study I've come across Morita contexts, but I'd like to understand them better. A quick Google search doesn't yield much fruit, so I was hoping to find a good reference on the ...
3
votes
1answer
463 views

Problem on a finite commutative ring with no zero divisors [duplicate]

This is a problem from Dummit & Foote. Prove that a non-zero finite commutative ring that has no divisor is a field. (Do not assume the ring has a 1) Evidently, one has to use the theorem ...
2
votes
1answer
91 views

Can the Euclidean algorithm fail by not terminating in non Euclidean domains?

Is it possible for the Euclidean algorithm to fail by not terminating in finite time in non-Euclidean domains? In $\mathbb{Z}[X]$ it can fail by going out of the ring, ie one gets a non integer ...
1
vote
3answers
171 views

Are there any zero divisors in this ring?

Definition: Zero-Divisors. A nonzero element $a$ in a commutative ring $R$ is called a zero divisor if there is a non zero element $b\in R$ such that $ab=0$. Consider the set $\mathbb Z$ ...
0
votes
2answers
85 views

how do we prove that ring of characteristic $p$ has arbitrarily large models?

As title says, how do we prove that the theory that describes ring of characteristic $p$ has arbitrarily large model? I am asking for a model-theoretic approach.
3
votes
4answers
251 views

Integral domains with non-trivial group of units that are not fields

I'm looking for examples of integral domains that are not fields but at the same time have more units than just the multiplicative identity 1. It's clear to me that by Wedderburn's little theorem, ...
1
vote
1answer
242 views

Finitely generated integral domain and finitely generated $k$-algebra.

Let $k$ be a field and $R$ a finitely generated domain over $k$. Then there are $y_1, \ldots, y_n$ such that $$R=k[y_1, \ldots, y_n]/P,$$ where $P$ is some prime ideal of $k[y_1, \ldots, y_n]$. My ...
3
votes
5answers
515 views

Can a ring of positive characteristic have infinite number of elements?

For curiosity: can a ring of positive characteristic ever have infinite number of distinct elements? (For example, in $\mathbb{Z}/7\mathbb{Z}$, there are really only seven elements.) We know that any ...
2
votes
0answers
71 views

A commutative ring with alternating and commutativity properties with infinite distinct elements

Is there any nontrivial commutative ring without multiplicative identity that satisfies alternating property ($x \cdot x = 0$ for all $x$ where $\cdot$ is multiplication operator and $x \cdot y \neq ...
1
vote
1answer
434 views

Let $I$ be a minimal ideal of a ring $R$. Then, $I$ is a direct summand of $R$ if and only if $I^2\neq \{0\}$.

Prove this statement: Let $I$ be a minimal ideal of a ring $R$. Then, $I$ is direct summand of $R$ if and only if $I^2\neq \{0\}$. (where direct summand means that we consider $I$ as a ...
2
votes
2answers
168 views

Two principal ideals coincide if and only if their generators are associated

Suppose we have a ring $R$ and $(a),(b)$ are both ideals of $R$. Is it always true that $(a)=(b)$ if and only if there exists a unit $c$ such that $a=bc$ (i.e., $a$ and $b$ are associate)? I ...
2
votes
1answer
111 views

Zero divisors in crossed group rings

It is not difficult to see that a group ring $K[G]$ ($K$ a domain) has non-trivial zero-divisors whenever there exists a non-trivial torsion element $g\in G$. [In fact, in this case, $1-g$ is such a ...
2
votes
3answers
824 views

Proof of “Freshman's dream” in commutative rings

When $p$ is a prime number and $x$ and $y$ are members of a commutative ring of characteristic $p$, then $$(x+y)^p=x^p+y^p.$$ This can be seen by examining the prime factors of the binomial ...
-1
votes
2answers
59 views

commutative ring that if two numbers are multiplied or added, product or sum can only be dissected into original two numbers

Is there any commutative ring that satisfies all of the following: 1) when two numbers are multiplied, the resulting product can only be dissected into these two numbers or their "prime factor" ...
1
vote
3answers
46 views

Can the characteristic of some ring differ from number to number?

The characteristic of ring is defined as the minimum times multiplicative identity must be added to get additive identity. My question is, for every number of ring, can the minimum times a number ...
1
vote
0answers
14 views

Are there ways to get modulated values similar to cyclic and negacyclic convolutions?

The Wikipedia article on the Schönhage–Strassen algorithm states that there are methods that can get values modulo $a^n+1$ or $a^n-1$ for some value $a$. More specifically, it shows that the cyclic ...
1
vote
0answers
42 views

Ring of holomorphic functoins

Let $\ O_n =\{\text{all holomorphic functions around the origin in} \Bbb C^n\}$, I'm trying to prove the follwoing, if $f_i=z^2-w^{n_i},i=1,2$. then$$\frac{O_2}{f_1}\simeq\frac{ O_2}{f_2}$$if and only ...
4
votes
1answer
60 views

Extending abelian groups to rings

I've been reading this article about extending abelian groups to rings: http://www.math.udel.edu/~coulter/papers/rings.pdf. Could you explain to me why theorem 2.1 guarantees left and right ...
1
vote
2answers
216 views

let $R$ be a ring , Show that $J(R/J(R))=\{0\}$.

$(a)$ let $R$ be a ring , Show that $J(R/J(R))=\{0\}$. $(b)$ Let $(R_i)_{i\in I}$ be a family of rings. Show that $J(\prod _{i\in I} R_i)= \prod _{i\in I} J(R_i). $ Definition: The Jacobson ...
1
vote
1answer
395 views

Every integral domain can be embedded in a field. Can this be generalized?

Every integral domain can be embedded in a field. I'm wondering if this result can be generalized. My question is, "Can every commutative unital ring $R$ be embedded in another ring $S$ such that the ...