2
votes
3answers
39 views

Example of ring $R$ with ideals $I\neq J$ such that $R/I \cong R/J$ as modules

It's easy to prove that if $I$, $J$ are two-sided ideals and $R/I\cong R/J$ as modules over $R$, then $I=J$. What about left ideals? Is there a simple counterexample? I believe I've found an answer, ...
0
votes
1answer
16 views

Counterexamples for the image of central,idempotent,invertible and nilpotent elements of a ring

I already proved that if i was given a surjective ring morphism f from R to S and then if a∈R is invertible, central, idempotent, or nilpotent, respectively then f(a) also is. But im looking for ...
0
votes
0answers
42 views

Example of a module over a non-abelian ring

Is there an example of a module over a non-commutative ring in which two maximal independet sets have distinct cardinality? Let $M$ be a module over a ring $R$. As usually a subset of ...
2
votes
2answers
55 views

Example of Localization and Prime Ideals

For each $n\in \mathbb Z^+$, give an example of a localization of $\mathbb Z$ with exactly $n$ prime ideals. Justify your answer. Could an example have something to do with a UFD or Noetherian ...
1
vote
1answer
47 views

Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

In the appendix of my commutative algebra text: Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$. ...
10
votes
3answers
107 views

An example of a noncommutative PID

It's well known that when a ring $R$ is a PID, every submodule of a free $R$-module is free. I'm interested in cases when the converse holds -- that is, in rings $R$ which have the property that every ...
17
votes
2answers
736 views

Proof that a certain subset of the reals is not a ring

Let $A = \{x \sin x : x \in \mathbb{Z}\} \subset \mathbb{R}$. Is $A$ a ring under the usual addition and multiplication operations of $\mathbb{R}$? It looks like it's not, but I can't find something ...
1
vote
3answers
166 views

Counterexamples to Nakayama's Lemma if $M$ is not finitely generated

One of the most famous forms of Nakayama's lemma says: Let $I$ be an ideal in $R$ and $M$ a finitely-generated $R$ module. If $IM = M$, then there exists an $r \in R$ with $r ≡ 1 \pmod I$, ...
0
votes
2answers
104 views

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$? [closed]

What is an example of a ring $A$ that is isomorphic to ${\Bbb R}^{5}$ ? Im sorry I am confused about ring theory. Its all new to me.
8
votes
2answers
75 views

Maximal ideals in $C^\infty(\mathbb{R})$

I know that for a compact manifold $M$ any maximal ideal in the algebra $C^\infty(M)$ of smooth functions on $M$ is of the form $\mathfrak{m}_p=\{f\in C^\infty(M)|f(p)=0\}$. For example, the proof is ...
6
votes
2answers
115 views

Do silly-rings exist?

A ring can be defined as a near-ring satisfying two-sided distributivity, whose underlying additive group is Abelian. Negating this second stipulation, we obtain the following definition. A silly-ring ...
2
votes
1answer
33 views

Does there exist a reversible monoid that fails to be Dedekind-finite?

Call a ring with unity reversible iff $xy = 0$ implies $yx = 0$. Dedekind-finite iff $xy = 1$ implies $yx = 1$. It is proved here that every reversible ring is Dedekind-finite. Now clearly, the ...
2
votes
2answers
57 views

A non-UFD where we have different lengths of irreducible factorizations?

A unique factorization domain (UFD) is defined to be an integral domain where each element other than units and zero has a factorization into irreducibles, and if we have two such factorizations then ...
4
votes
1answer
184 views

Example of rings of the same positive characteristic that do not embed into their tensor product?

I'm overcoming my fear of tensor products, and the following exercise got me wondering: Give an example of commutative rings $A$ and $B$ with $\operatorname{char}A=\operatorname{char}B$ such that ...
1
vote
1answer
63 views

Example of a ring $R$ and ideals $A$ and $B$ of $R$ such that $AB\neq A\cap B$.

Give an example ofa ring $R$ and ideals $A$ and $B$ of $R$ such that $AB\neq A\cap B$. I know that $AB\subset A\cap B$. But I can't find an example such that $AB\neq A\cap B$.
2
votes
1answer
82 views

Integral extensions of rings, when one of the rings is a field

The following is from page 61 of Introduction to Commutative Algebra by Atiyah & Macdonald: Proposition 5.7. Let $A\subseteq B$ be integral domains, $B$ is integral over $A$. Then $B$ is a ...
1
vote
2answers
125 views

Must certain rings be isomorphic to $\mathbb{Z}[\sqrt{a}]$ for some $a$

Consider the group $(\mathbb{Z}\times\mathbb{Z},+)$, where $(a,b)+(c,d)=(a+c,b+d)$. Let $\times$ be any binary operation on $\mathbb{Z}\times\mathbb{Z}$ such that ...
1
vote
3answers
104 views

Ring and Subring with different Identities [duplicate]

Is there an example of a ring $S$ with identity $1_S$ containing a non-trivial subring $R$ which itself has an identity $1_R$, but $1_R\neq 1_S$ (or equivalently $1_S\notin R$). I'd also like to know ...
3
votes
4answers
113 views

A domain with only a (non-zero) prime ideal

What is an example of a domain $A$ such that Spec$A=\{(0),\mathfrak p\}$? For instance one could find a principal ideal domain that is also a local ring but I can't imagine such a ring.
14
votes
3answers
467 views

Are there interesting rings without unity?

There are several introductory textbooks which define a ring without any reference to a unity. However, nearly all of the rings one encounters in various branches of mathematics are endowed with a ...
1
vote
2answers
106 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 ...
3
votes
4answers
182 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, ...
5
votes
2answers
264 views

Injectivity of Homomorphism in Localization

Let $\alpha:A\to B$ be a ring homomorphism, $Q\subset B$ a prime ideal, $P=\alpha^{-1}Q\subset A$ a prime ideal. Consider the natural map $\alpha_Q:A_P\to B_Q$ defined by ...
0
votes
0answers
79 views

Complete intersection but not a domain

This may be a rather trivial question but here it goes. I am looking for a scheme defined by (more than one irreducible and reduced) homogeneous equation in a polynomial ring that is a complete ...
17
votes
2answers
725 views

A (non-artificial) example of a ring without maximal ideals

As a brief overview of the below, I am asking for: An example of a ring with no maximal ideals that is not a zero ring. A proof (or counterexample) that $R:=C_0(\mathbb{R})/C_c(\mathbb{R})$ is a ...
6
votes
1answer
226 views

If $R$ is commutative, and $J\lhd I\lhd R,$ does it follow that $J\lhd R?$

$\lhd$ will stand for "is an ideal of" in this post. Let $R$ be a commutative ring, $J\lhd I\lhd R$. Does it follow that $J\lhd R?$ I don't think it does, but I'm having difficulty finding a ...
5
votes
1answer
306 views

Non-Noetherian rings with an ideal not containing a product of prime ideals

It is well-known that in every commutative Noetherian ring every ideal contains a product of prime ideals. Are there examples of non-Noetherian rings with an ideal that does not contain any prime ...
4
votes
2answers
397 views

Left and right ideals of $R=\left\{\bigl(\begin{smallmatrix}a&b\\0&c \end{smallmatrix}\bigr) : a\in\mathbb Z, \ b,c\in\mathbb Q\right\}$

If $$R=\left\{ \begin{pmatrix} a &b\\ 0 & c \end{pmatrix} \ : \ a \in \mathbb{Z}, \ b,c \in \mathbb{Q}\right\} $$ under usual addition and multiplication, then what are the left and right ...
1
vote
2answers
152 views

If a sub-C*-algebra does not contain the unit, is it contained in a proper ideal?

Suppose that $R$ and $S$ are unital rings and that $S$ is a subring of $R$ in the weak sense where the multiplicative identities $1_R$ and $1_S$ are not assumed to be the same. In fact, assume $1_R ...
28
votes
1answer
2k views

An example of a division ring $D$ that is **not** isomorphic to its opposite ring

I recall reading in an abstract algebra text two years ago (when I had the pleasure to learn this beautiful subject) that there exists a division ring $D$ that is not isomorphic to its opposite ring. ...
3
votes
0answers
87 views

Azumaya algebra and its subalgebras

I remind you that an Azumaya algebra $A$ is a central and separable algebra. Now, I know that if $A$ is an algebra over a skew-field or over a local ring then there exists a subalgebra $S$ of $A$ such ...
8
votes
5answers
1k views

Non-unital rings: a few examples

Every ring I've ever heard of is unital, i. e., contains a (unique) element $a$ such that $xa = ax = x$ for every $x$ in it. However, some rings do not have such an element. What are they? P. S.: one ...
3
votes
6answers
2k views

Examples of rings with idempotent elements

As a part of my studies in ring theory, I've encountered the concept of an idempotent element, i.e., an element $e$ such that $e^2=e$. Are there some interesting examples of rings with idempotent ...