This tag is for questions about rings, which are a type of structure studied in abstract algebra and algebraic number theory.
2
votes
2answers
52 views
Are there analogues of eigenvalues/eigenvectors for a ring homomorphism/endomorphism?
My question is very simple. To put it in a context, a linear transformation is nothing but a homomorphism from a vector space to another. I usually visualize the action of a linear transformation by ...
0
votes
0answers
29 views
What is 0 mapped to in a Euclidean domain?
Let us suppose we have a Euclidean domain A, in which we have $a=q*d+r$. We know that there is a function $f:A\to Z$ such that for every $a\in A/0$, we have $f(a)>f(0)$. Also, $f(r)<f(d)$.
Is ...
0
votes
0answers
12 views
Finite generation of a subalgebra of R[x]
Let $k$ be a field, $R$ a noetherian $k$-domain, $x$ a central variable over $R$ and $A$ a (commutative) $k$-subalgebra of $R[x]$. So $R[x]$ is an Ore domain. Now, suppose that there exists a finitely ...
1
vote
1answer
57 views
Field extension of $\mathbb Q$ of degree 2
Assume $K:\mathbb Q$ is a field extension and $[K:\mathbb Q] = 2$. Show that there is a unique squarefree $d \in \mathbb Z$ such that $K = \mathbb Q(\sqrt d)$.
I know that $K$ is generated by say ...
6
votes
0answers
73 views
An example of a compact multiplicatively unbounded ring
My teacher asked me to build an associative topological Hausdorff compact ring $R$ with $1$, which is multiplicatively unbounded. That means there is a neighborhood $U\ni 1$ such that $FU\not=R$ for ...
1
vote
1answer
38 views
Do ideals partition a ring?
Say we have two principal ideals- $(a)$ and $(b)$. Is $r_{1}*a=r_{2}*b$ possible for $r_{1},r_{2}\in R$, with $(a) \neq (b)$?
I don't see a problem with this as long as the multiplicative inverses of ...
1
vote
0answers
54 views
What is $\mathbb{C}[xy]/\langle x\rangle \subseteq \mathbb{C}[x,y]/\langle x \rangle$?
Consider the ring $\mathbb{C}[x,y]$,
and consider
$$R=\dfrac{\mathbb{C}[xy]}{\langle x\rangle } \subseteq \dfrac{\mathbb{C}[x,y]}{\langle x\rangle }\cong \mathbb{C}[y].$$
Is $R\cong ...
3
votes
1answer
51 views
Correspondence between submodules and quotient modules
What is the (natural) bijection between the set of all sub modules upto isomorphism and set of all isomorphic quotient modules upto isomorphism of a finitely generated torsion module over a PID. Is ...
2
votes
1answer
72 views
Prove that $D[x]$ is an integral domain if $D$ is one.
Prove if $D$ is an integral domain and $f,g\in D[X]$ are nonzero, then $fg$ does not equal $0$ and $\deg[f(x)g(x)]=\deg f(x) + \deg g(x)$.
I do not know much about this since I just learned about it. ...
2
votes
0answers
37 views
Modules with maximal submodules and projective dimension
If $R$ is a left noetherian ring, then every finitely generated left $R$-module $M$ is noetherian, and hence every proper submodule of $M$ is contained in some maximal submodule of $M$.
Is it ...
3
votes
3answers
136 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
32 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
51 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 ...
7
votes
1answer
67 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 ...
0
votes
1answer
66 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
67 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
94 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
51 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
58 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
92 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
65 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
44 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
67 views
What is “prime factorisation” of polynomials?
I have the following question:
Find the prime factorisation in $\mathbb{Z}[x]$ of $x^3 - 1, x^4 - 1, x^6 - 1$ and $x^{12} - 1$. You will need to check the irreduciblity in $\mathbb{Z}[x]$, of ...
2
votes
1answer
52 views
Subring of Z[x] generated by set of integers and polynomials
Let Z be the ring of integers. We have the subring of Z[x] generated by integers and p1 and p2 (p1 and p2 are polynomials over Z, we note it as Z[p1,p2]). I've got for my homework to investigate if ...
1
vote
3answers
86 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 ...
1
vote
6answers
139 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 ...
1
vote
1answer
108 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
40 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
37 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
50 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
44 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
$ ...
1
vote
2answers
72 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
26 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
29 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.
1
vote
2answers
27 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
43 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
44 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
25 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 ...
21
votes
0answers
389 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 nontrivial ...
3
votes
1answer
48 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
32 views
Proving an ideal is principal
Let $R,\mathfrak{P},\overline{\mathfrak{P}}$ and $p$ be as in this question. I have proved that $\mathfrak{P}\cdot\overline{\mathfrak{P}}=pR$. I think this can be used for proving what follows, by I ...
0
votes
0answers
39 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 ...
3
votes
1answer
27 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
107 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
45 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 ...
0
votes
3answers
71 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
74 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
61 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
33 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
164 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 ...
