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Questions tagged [rngs]

A rng is an associative ring without necessarily having a multiplicative identity (rng = ring - $i$dentity).

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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 $A\subseteq P$ ...
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1answer
35 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
121 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
89 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
51 views

$\mathrm{ann}(R/I) = I$ 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
159 views

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
53 views

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
75 views

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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2answers
138 views

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
622 views

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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3answers
171 views

$R$ 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
31 views

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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2answers
177 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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Examples of Finite Non-Unital Integral Rings [duplicate]

Here, integral rings are rings without nonzero zero-divisors. In this question, rings are not assumed to be unital (i.e., they may not have the multiplicative identity). Some people call them rngs. ...
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2answers
250 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
416 views

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
51 views

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
281 views

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 $$ eRe \oplus (1-e)Re \oplus eR(1-e) \oplus (1-e)R(1-e). $$ I tried to construct ...
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2answers
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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
513 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
461 views

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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2answers
117 views

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
357 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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1answer
132 views

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
138 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
140 views

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 ...
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180 views

Element of a ring without unity which divides every other element

Question. Is there an example of a ring $R$ (commutative or not) without unity and an element $x \in R$ such that for every $y \in R$ there exists a $z \in R$ such that $y = x z$? In other words, is ...
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1answer
295 views

Prove that every maximal ideal of a commutative ring $R$ with $R^2=R$ is prime

Prove that every maximal ideal of a commutative ring $R$ (not assumed to have $1$) with $R^2=R$ is prime. If $M$ is a maximal ideal of $R$, I am trying to prove that for all $a,b,ab \in M$ implies $a\...
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111 views

algebras without identity

This problem is an exercise from Drozd-Kirichenko's book Finite Dimensional Algebras, page 29. Let $k$ be a field. Let $A$ be a $k$-algebra not necessarily with identity. Let $\overline A$ be the ...
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1answer
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Rings and Rngs: properties which differ depending on the inclusion of a multiplicative identity.

As far as I know, one can define a ring with or without a multiplicative identity. My question is: what kind of properties, theorems, etc. get lost when one talks about rngs instead of rings, and ...
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1answer
845 views

Determine if R is a commutative ring with unity?

On the set $R-\{-1\}$ define the operations $a\oplus b = a + b + ab$ and $a \times b = 0$. Determine if $\big(R-\{-1\}, \oplus,\times\big)$ is a ring. Is it a commutative ring with unity? Using the ...
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1answer
312 views

If R is a commutative ring, but has no identity is c and element of I?

Problem statement: Let $c \in R$ and let $ I = \{rc\mathrel{|}r\in R\} $. If $R$ is commutative but has no identity, is $c$ an element of the ideal $I$? Proof: Suppose $c \notin I$ and $R$ is a ...
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2answers
361 views

Nontrivial subring with identity of a ring without identity [duplicate]

I'm looking for an example a ring and a subring with $R \subset S$ such that $R$ has 1 but $S$ does not. Its easy to choose R to be the trivial ring with $0=1$, but are there any more exotic examples ...
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1answer
74 views

PRNG for compression

I'm trying to intuitively grasp information theory. You have a string of size X that contains a lot of information, say it's a movie. You have a string of size N << X which is going to be the ...
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defining gcd on rings

I see that in most textbooks they say let $R$ be an integral domain and start defining the greatest common divisor. My question is, can gcd's be defined on just commutative rings without an identity?
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Must this rng be a ring?

A rng is a ring without the assumption that the ring contains an identity. Consider a finite rng $\mathbf{R}$. I am investigating conditions that get close forcing an identity but not quite. The ...
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Free $R$-module when $R$ is not unital

We can easily construct free $R$-module when $R$ is unital by setting $$R[S] = \{ f\colon S\to R\,|\, f\ \text{finitely supported}\}$$ and defining operations pointwise. The key here is that we can ...
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For $\operatorname{char}(R)=\bar n$ and $\bar m<\bar n$, show that $\bar m\cdot x=0$ is only possible in a “trivial” manner.

Define $\operatorname{char}(R)$ as the least positive integer $\bar n$ for which: $\bar n\cdot x=\underbrace{x+x+\ldots+x}_{\bar n\text{ times}}=0$ for all $x\in R$. We say $\bar n=0$ when no positive ...
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Bounding the number of commutative rings with identity of size n

For integer $n \geq 2$, the number of rngs of size $n$ is in general an open problem. Let $a(n)$ be the number of commutative rngs of size $n$. We can use the facts that $a(n)$ is multiplicative and ...
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1answer
129 views

Ring of even integers considered as module over itself

I wonder, if the ring without unity $2\mathbb{Z}$, considered as a modul over itself, is a free modul. For a ring with unity, which is not the nullring the answer is clearly yes, because one can ...
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10k views

In a ring, how do we prove that a * 0 = 0?

In a ring, I was trying to prove that for all $a$, $a0 = 0$. But I found that this depended on a lemma, that is, for all $a$ and $b$, $a(-b) = -ab = (-a)b$. I am wondering how to prove these ...
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2answers
119 views

Prove that every rng with a left-side identity has a right-side identity?

Given a rng $(R,+,\cdot)$ and $e \in R$ with $\forall a \in R: e \cdot a = a$, I would like to prove $a \cdot e = a$. What I have so far is the following: Assume there exists $r \in R$ with $\forall ...
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4answers
3k views

Why is it necessary for a ring to have multiplicative identity?

I have read earlier that in a ring $(R,+,.)$ the following needs to hold: $(R,+)$ is an abelian group multiplication is associative and closed left and right distribution laws hold. However, I ...
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1answer
337 views

Does $M \otimes_R N = 0$ for a non-unital ring $R$ if there are ideals $I,J \lhd R$ such that $MI+JN = 0$ and $I+J = R$?

Suppose that $M,N$ are $R$-modules, $I,J\lhd R$. Suppose that $MI=JN=0$ and that $I+J=R$. Prove that $M\bigotimes_R N=0$. This is very easy to prove if the ring is unital as you may write $1=i+j$, ...
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1answer
245 views

Are there different left and right ideals in a ring without identity?

For a non commutative ring without identity, is it possible that there will be right and left ideals which are different?
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4answers
1k views

Pathologies in “rng”

There is no general consensus regarding whether a ring should have a unity element or not. Many authors work with unital rings , and other does not essentially require unity. If we do not assume ...
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2answers
164 views

For any rng $R$, can we attach a unity?

Let $R$ be an rng. (There may be no unity) Then, does there always exist a ring(with unity) $A$ such that $R$ is a subrng of $A$?