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
112 views

For a group-algebra $k[G]$ ($G$ finite), why is a $k[G]$-module the same as a $k$-representation of $G$?

I'm reading the Atiyah-MacDonald book on Commutative Algebra. At the beginning of the module chapter on page 17, they make an example which I don't understand. Example 5) is: $G$ = finite group, ...
1
vote
0answers
85 views

Another problem with a proof in “Topics in Algebra” by Herstein- every element in R can be written as a finite product of prime numbers.

On pg. 146 of the second edition, it says let $a=bc$, where $a,b,c\in R$. $R$ is a Euclidean ring. If $d:R\to Z$, then if $b$ is a unit, $d(a)=d(c)$ [$d(a)=d(b)$ if $c$ is a unit]. If neither $b$ nor ...
1
vote
1answer
360 views

Problem with proof in “Topics in Algebra” by Herstein.

On pg. 148 of the second edition, the proof for the unique factorization theorem in Euclidean rings is given. Let $\pi_{1}.\pi_{2}.\pi_{3}\dots = \pi_{1}'.\pi_{2}'.\pi_{3}'\dots\dots$, where all the ...
0
votes
1answer
99 views

Showing that the ring of $n\times n$ matrices has exactly two 2 sided ideals, even though it is not a division ring

Show the ring $A=\mathrm{Mat}(F)$ has exactly two 2-sided ideals, even though it is not a division ring. $F$ is any field, and $\mathrm{Mat}(F)$ is all $n\times n$ matrices with elements of $F$ ...
1
vote
2answers
224 views

Find all maximal ideals of $\mathbb{Z}_{540}$

Find all maximal ideals of $\mathbb{Z}_{540}$ By using the following statement, '$f:R \rightarrow S$ be a surjective ring homomorphism and let $K=ker(f)$. Observe that there is a one-to-one ...
1
vote
2answers
133 views

Examples of $F$-algebra which has no proper two-sided ideals but has one-sided ideal(s)

Let $F$ be a field and $A$ an $F$-algebra. (And assume that $A$ is finite dimension over $F$ if necessary.) A textbook says that $A$ is simple if it has no proper two-sided ideals. To understand this ...
8
votes
1answer
128 views

Commutativity characterization?

Let $R$ be a ring (not necessarily unital) and for any $x\in R$ there is an integer $n \geq 2$ s.t. $x=x^2+\cdots+x^n.$ Does it imply that $R$ is commutative?
6
votes
4answers
855 views

How to find all the maximal ideals of $\mathbb Z_n?$

How to find all the maximal ideals of $\mathbb Z_n?$ I think $(0)$ is the only maximal ideal of $\mathbb Z_n$ for if $a$ is a non-unit in a maximal ideal of $\mathbb Z_n$ then ...
1
vote
1answer
291 views

Krull Dimension

I'm studying Krull dimension and I'm confused about the definition of $\text{ht}(P)$, which is as I understand is the following: let $$P_0\subset P_1\subset\dots\subset P_n=P$$ be a chain of prime ...
2
votes
2answers
110 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
45 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 ...
2
votes
1answer
196 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 ...
7
votes
0answers
118 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
72 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
74 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
255 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 ...
3
votes
1answer
600 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. ...
4
votes
1answer
148 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 ...
7
votes
3answers
1k 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
83 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
111 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
721 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
125 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
132 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
86 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
81 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
592 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
103 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
64 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
935 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
199 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
466 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
939 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
62 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
71 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 ...
4
votes
1answer
105 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
50 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
135 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
31 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
43 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
86 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 ...
156
votes
0answers
7k 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
56 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
58 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
69 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
513 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 ...