Questions tagged [rngs]
A rng is an associative ring without necessarily having a multiplicative identity (rng = ring - $i$dentity).
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Looking for counterexamples (non-domains)
I stumbled upon this question Why doesn't $xa = x$ for all $x \in R$ imply that $a$ is the unit of $R$? and understood the given answers. But it got me thinking about a counterexample, and I was ...
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If ring $R\neq \{0\}$ then subring $R_0 \neq \{0\}$ (proof verification)
I want to check my proof:
Given: Let $R$ be a ring with $1$ and $H$ an additive subgroup of $R$. We define $R_0=\{x\in R : \forall h \in H \text{ we have } xh \in H\} \subset R$.
To prove: $R\neq \{0\}...
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$R$ a non-unitary ring such that $R^+ \cong \mathbb{Q}/\mathbb{Z}$
We are given $R$ a non-unitary ring such that $R^+ \cong \mathbb{Q}/\mathbb{Z}$. Prove: $ab=0$ for all $a,b \in R$.
Here is what I have attempted so far:
We take two arbitrary elements of $\mathbb{Q}/\...
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How can you prove Nakayama's lemma over nonunital rings using the unitization?
For a nonunital commutative ring $A$, an $A$-module $M$ is called finitely generated over $A$ if there is a finite set of elements $x_1,...,x_n\in M$ such that every element of $M$ is of the form $...
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Can the union of an ascending sequence of ideals of $R$ be $R$?
I am confused about the equivalence of the different definitions of Noetherian Rings. The confusion essentially stems from the fact that a Noetherian ring need not be finitely generated. Essentially ...
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1answer
63 views
Structural differences between $2 \mathbb{Z}$, $3 \mathbb{Z}$ as rings. [duplicate]
Is usual to find in abstract algebra books this exercise:
Show that $2 \mathbb{Z}$ and $3 \mathbb{Z}$ are isomorphic as groups (with usual sum) but they aren't as rings (with usual sum and addition).
...
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In a finite commutative ring , every prime ideal is maximal?
I am stuck in a true/false question. It is
In a finite commutative ring, every prime ideal is maximal.
The answer says it's false.
Well what I can say is (Supposing the answer is right)
$(1)$ The ring ...
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2answers
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Nontrivial subring of $\mathbb{R}$ not containing $1$
Are there examples of nontrivial subrings of $\mathbb{R}$ that do not contain $1$? If not, how can we prove they don't exist? The definition of "ring" here is really "rng"; rings ...
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1answer
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Simple example for a rng with an inverse semi-group as the multiplicative group
I’m looking for a ring without an multiplicative identity, and in which every element $x$ has a weak inverse $y$ such that $xyx=x,yxy=y$ preferably simple to construct and or of finite size.
If it has ...
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1answer
94 views
Characterization of Injective rings homomorphism from $\mathbb{Z}_m$ to $\mathbb{Z}_n$
Show that there is an injective ring homomorphism $f:\mathbb{Z}_m \rightarrow \mathbb{Z}_n$ if and only if $m\mid n$ and $\frac{n}{m}$ is relatively prime with $m$.
In one direction, was not ...
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Is the ring $3\mathbb Z$ a ring homomorphic image of the ring $2\mathbb Z$.
Is the ring $2\mathbb Z$ isomorphic to the ring $3\mathbb Z$ ?
Solution: Let if possible,$\phi:\mathbb {2Z\to 3Z}$ be a ring isomorphism.
Then $\phi$ is a group isomorphism between the additive ...
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Given prime ideal $P$ of ring $R$, does $RrR\subset P$ imply $r\in P$ for $r\in R$? (Hungerford)
I am trying to understand a problem in Hungerford (p. 134, Exercise 14). Specifically, it says that if $P$ is a prime ideal in a not necessarily commutative ring $R$, and $r,s\in R$ such that $rRs\...
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Sum of principal ideals in a commutative rng
Let's define the principal ideal of an element $a$ of a commutative ring $R$ with or without identity as $\langle a \rangle = R \cdot a + \mathbb Za$.
It looks like with this definition $\langle a \...
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1answer
115 views
Maximal non-unit ideal in a ring with or without identity
Every maximal ideal is prime in a commutative ring with identity.
There were several posts on the site about analogues of the claim for rngs (rings with or without identity):
A maximal ideal is ...
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Product of classes of associates in a commutative ring/rng
Let $[a]$ be the equivalence class of associates of an element $a$ of a ring/rng.
Two elements of a ring/rng are associates ($\sim$) if they generate the same principal ideal.
$[a] \cdot [b] \sim [...
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Semiring of classes of associates of a ring/rng
Let's call elements that generate the same principal ideal of a ring/rng associates.
$[a]$ is the equivalence class of associates of an element $a$ of a ring/rng.
Let $A = \{[a],[b], ...\}$ be the ...
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1answer
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In a non-unitary commutative ring, every maximal ideal is primary?
Let $R$ be a commutative ring without identity.
My question: is it true or false that every maximal ideal of $R$ is primary?
(An ideal I of R is said primary if is proper and
$\forall a,b\in R, ab\...
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Factorization in a principal ideal ring/rng
It is known that every PID is a UFD.
Is it true that every element of a commutative principal ideal ring (PIR) or rng that is not zero and not a unit is a product of a finite number of irreducible ...
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Element of a ring acting as a permutation on an ideal
I am investigating cases when $r \cdot I = I$ for some element $r$ and an ideal $I$ of a commutative ring or rng R.
Clearly, $r \cdot \langle 0 \rangle = \langle 0 \rangle$ for any element $r$ of $R$,...
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In a commutative rng with comaximal ideals, product equals intersection as well?
In a commutative ring $R$ with unity, the product of every two comaximal ideals equals their intersection, that is, if $I + J = R$, then $I\cap J = IJ$.
The proof I know involves the unity of $R$, so ...
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1answer
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Example of two subrings with unity of a ring with unity whose intersection is non trivial and has no unity.
I was just thinking about the intersection of rings and this question popped up, I tried giving an example and proving that the intersection had to have an unity, but was unsuccessful in both.
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Can a ring have no zero divisors while being non-commutative and having no unity?
I was wondering if, in a ring, the property of having no zero-divisors (except for zero itself) is independent from the ring being commutative or from having a unity (i.e.multiplicative identity) so I ...
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Prime property in noncommutative rings without identity
Let $R$ be a ring (without assuming identity or commutativity), and $P$ a proper ideal of $R$. Show that the following are equivalent:
(a) For ideals $A,B$: $AB\subseteq P$ implies $A\subseteq P$ or $...
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1answer
57 views
In a commutative ring $R$ if, for every $a\in R$, the smallest ideal containing $a$ is equal to $Ra$ then $R$ has identity?
Let $R$ be a commutative ring such that, for every $a\in R$, the smallest ideal containing $a$ is equal to $Ra$. Does $R$ have identity?
I did this when $R$ has at least one non-zero divisor then it'...
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1answer
381 views
Non-unital ring $(2\mathbb{Z})[X]$ is not Noetherian
Let $R = 2\mathbb{Z}$. Then $R[x]$ is not a noetherian ring.
I do not understand why this is so, because Hilbert's basis theorem says: If R Noetherian ring, then R[X] a is Noetherian ring (from wiki)....
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1answer
170 views
A non-Boolean ring without unity with this property
I'm exploring rings that have the following property:
$\rule{10cm}{0.4pt}$
For all $x \in R$, and for all $n \in \mathbb{N}$, there exists $y \in R$ such that $\sum_{i=1}^{n} x = y \cdot x$.
$\rule{...
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1answer
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$\mathrm{ann}(R/I) = I$ only when $R$ is unital?
My professor wrote down that $\sqrt{\mathrm{ann}_R(R/I)} = \sqrt{I}$ for $I$ a proper ideal of $R$. Clearly if $R$ is unital ring we have that $\mathrm{ann}_R(R/I) = I$. So the radical relation is ...
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1answer
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Runs-Up/Down RNG test question.
There is not much relevant information to be found about Runs-Up/Down test on the great web. All I find is more or less just recycling the info than can be found in Knuth, The Art of Computer ...
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1answer
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Conditions on The Lemma 3.5.1 of Herstein
The lemma states that "Let R be a commutative ring with unit element whose only ideals are (0) and R itself. Then R is a field."
My question is that what if we don't assume the ring has unit element? ...
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1answer
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Definition: rng ideal versus algebra ideal in a non-unital $C^*$-algebra
When I first moved into operator algebras, equipped with the basics of abstract algebra, I encountered the following definition of ideals:
Definition (algebra ideals). A left (respectively, right) ...
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Every commutative ring of characteristic $p$ contains $\mathbb F_p$ as a subring?
I know that if a commutative ring with unity is of characteristic $p$ then it will contain $\mathbb F_p$ as a subring, but if the ring is commutative with characteristic $p$ and without unity then is ...
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1answer
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Ideal of direct sum of rings.
It is known that if $R_1$ and $R_2$ are rings with unity, then every ideal of $R_1 \oplus R_2$ has the form $I_1 \oplus I_2$, where $I_1$ and $I_2$are ideals of $R_1$ and $R_2$ respectively. ...
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$R$ a ring with 1 and charac. $n>1$ (resp. 0) $\implies R$ contains a subring $\simeq \Bbb{Z}_n$ (resp. $\Bbb{Z}$). What happens if $R$ has no 1?!
Definition: Let $R$ be a ring. We say that $n\in \Bbb{Z}_+$ is the characteristic of $R$ if it is the least positive integer such that $n r=0$, for all $r\in R$ (here $nr$ denotes $r+r+\dots+r$, "$n$ ...
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1answer
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If an ordered rng $A$ has a upper or lower bound, then $A = {0}$
Let $m$ be the upper bound of $A$. Then, $a \leq m$ for all $a\in A $.
Now I should maybe find a way to show that that inequality is in fact an equality. Or maybe there's an easier way that I don't ...
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337 views
$R$ is a prime right Goldie ring which contains a minimal right ideal. Show that $R$ must be a simple Artinian ring.
$R$ (1 is not assumed to be in $R$) is a prime right Goldie ring (finite dimensional and ACC on right annihilators) which contains a minimal right ideal. Show that $R$ must be a simple Artinian ring.
...
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2answers
359 views
Product of Principal Ideals when $R$ is commutative, but not necessarily unital
When $R$ is a ring (not necessarily commutative, and not necessarily with unity), I have a result that tells me that for $x \in R$, the ideal generated by $x$, $(x) $, is $= I_{x} = \langle RxR \...
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1answer
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Smallest subring of $\mathbb Q$ containing $3/10$
Let $R$ be the smallest subring of $\mathbb Q$ (the field of rational numbers) that contains $3/10$ ($R$ doesn't have to be a unital ring). Does $1 \in R$?
Is the desired smallest subring this one: $...
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1answer
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Does $\operatorname{End}_R(R)\cong R^{\text{op}}$ still hold when $R$ doesn't have an identity?
When I try to prove that $\operatorname{End}_R(R):=\text{Hom}_R(R,R)\text{(left)}\cong R^{\text{op}}$, I used the map $f \mapsto f(1)$, which requires $R$ to have an identity. Is it really necessary ...
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1answer
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Pierce decomposition of a (not necessarily commutative) ring and the cartesian product
Let $R$ be a (not necessarily) commutative ring and $e \in R$ some idempotent. Then the Pierce decomposition writes
$$
R=eRe \oplus (1-e)Re \oplus eR(1-e) \oplus (1-e)R(1-e).
$$
I tried to construct ...
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In the ring $6\mathbb{Z}$ is $12\mathbb{Z}$ maximal ideal but not prime ideal?
I need to prove that in the ring $6\mathbb{Z} = \left\{x \in \mathbb{Z} \mid x = 6q, q \in \mathbb{Z}\right\}$ the subset $12\mathbb{Z}$ is a maximal ideal but not a prime ideal.
I first wanted to ...
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1answer
629 views
Zero divisor in ring without unity [closed]
Let $R$ be a commutative ring without unity and $n \in R\setminus\{0\}$. Prove that $n\mid n$ implies that $n$ is a zero divisor.
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Is $2\mathbb{Z}$ ring isomorphic to $4\mathbb{Z}$?
A ring isomorphism $f: R \rightarrow S$ satisfies these properties:
a) $f(a + b)$ = $f(a) + f(b)$, for all $a, b \in R$.
b) $f(ab) = f(a)f(b)$, for all $a, b \in R$.
I'm inclined to believe that ...
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1answer
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Show that 3Z is not isomorphic to 5Z (when dealing with rings)
Show that the ring $3\mathbb Z$ is not isomorphic to the ring $5\mathbb Z$.
I see that they are not but I am not sure how to go about proving it. We went over a similar problem, disproving it by ...
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An interesting problem about a “quasi ideal”
Let R be an infinite ring without an unity. Let I ⊆ R satisfying: for any a,b ∈ I, r ∈ R, we have a+b ∈ I, ar ∈ I, ra ∈ I. Then there are two possibilities about I:
(1) I must be an ideal, i.e., for ...
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Ideals of the unitalization
We know that every commutative ring can be embedded in a ring with identity as follow:
Let $R$ be ring and $R_1=R\times \mathbb{Z}=\{(r,n)\mid r\in R,n\in \mathbb Z\}$. This is a ring with addition ...
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1answer
859 views
Ring (without identity) of characteristic $n$ contains elements of any order $k$ dividing $n$
Let $R$ be an arbitrary ring (i.e., may not have unity) with characteristic $n\in\mathbb{N}$ (positive integer) i.e any element added to itself $n$ times always yields additive identity $0$.
Show ...
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Is there a ring without unity which can be expressed as a union of its three proper ideals? [closed]
Are there any rings without unity which can be expressed as a union of its three proper ideals?
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Characteristic of a non unital integral domain [duplicate]
Theorem: the characteristic of a non unital integral domain must be $0$ or a prime.
Assume that $mn$ is the characteristic of a the integral domain
for every $a$ in the integral domain we have $(mn)a^...
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2answers
279 views
Example that $R/I$ is not field where $R$ is a commutative ring and $I$ is maximal ideal.
Theorem.
Let $I$ be an ideal in a commutative ring $R$ with identity. Then $I$ is a maximal ideal if and only if the quotient ring $R/I$ is a field.
Above Theorem is very famous theorem. But The ...
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2answers
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Is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$?
I came over a question in ring theory which I am not being able to proceed upon:
When is the ring $m\mathbb{Z}$ isomorphic to the ring $n\mathbb{Z}$, where $m, n \in \mathbb{N}$?
I know that to ...
