Questions tagged [rngs]
A rng is an associative ring without necessarily having a multiplicative identity (rng = ring - $i$dentity).
2
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0answers
28 views
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 $A\subseteq P$ ...
1
vote
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'...
3
votes
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)....
2
votes
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{...
3
votes
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 ...
1
vote
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 ...
0
votes
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? ...
4
votes
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) ...
3
votes
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 ...
2
votes
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. ...
3
votes
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 $r+r+\dots+r$, "$n$ ...
2
votes
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 ...
3
votes
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.
...
1
vote
0answers
34 views
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.
...
1
vote
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 \...
1
vote
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: $...
0
votes
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 ...
4
votes
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 ...
0
votes
2answers
150 views
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 ...
0
votes
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.
1
vote
6answers
669 views
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 ...
0
votes
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 ...
1
vote
0answers
46 views
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 ...
3
votes
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 ...
3
votes
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 ...
-1
votes
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?
0
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0answers
19 views
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^...
3
votes
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 ...
0
votes
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 ...
4
votes
2answers
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 ...
0
votes
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\...
1
vote
0answers
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 ...
3
votes
1answer
54 views
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 ...
1
vote
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 ...
1
vote
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 ...
4
votes
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 ...
0
votes
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 ...
2
votes
0answers
150 views
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?
14
votes
2answers
225 views
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 ...
1
vote
0answers
134 views
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 ...
3
votes
0answers
85 views
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 ...
1
vote
0answers
36 views
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 ...
0
votes
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 ...
4
votes
3answers
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 ...
2
votes
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 ...
13
votes
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 ...
4
votes
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 $1=i+j$, ...
2
votes
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?
17
votes
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 ...
8
votes
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 $A$?
