Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [rngs]

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

8
votes
1answer
4k views

If $I+J=R$, where $R$ is a commutative rng, prove that $IJ=I\cap J$.

So I basically have to prove what is on the title. Given $R$ a commutative rng (a ring that might not contain a $1$), with the property that $I+J=R$, (where $I$ and $J$ are ideals) we have to ...
2
votes
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 $A\subseteq P$ ...
3
votes
1answer
122 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)....
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 ...
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 ...
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'...
10
votes
5answers
4k 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 ...
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 ...
9
votes
2answers
2k views

the ring of dual numbers over a field $k$

Suppose $k$ is a field,then the quotient ring $k[\epsilon]/\epsilon^2$ is called the ring of dual numbers over $k$. I learn this from Hartshorne. I wonder why it has this name(maybe this question is a ...
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{...
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
623 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. ...
26
votes
3answers
7k views

A maximal ideal is always a prime ideal?

A maximal ideal is always a prime ideal, and the quotient ring is always a field. In general, not all prime ideals are maximal. 1 In $2\mathbb{Z}$, $4 \mathbb{Z} $ is a maximal ideal. ...
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 \...
8
votes
2answers
237 views

Books on Rings without Identity

I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's Algebra textbook ...
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 ...
2
votes
1answer
261 views

$R$ be a ring without identity. If $R$ has a maximal left ideal, then the Jacobson radical is still the intersection of all the maximal left ideals?

We know that the definition of the Jacobson radical $J(R)$ (a) in a ring $R$ with identity is: $$J(R)=\cap \mbox{ maximal left ideals}.$$ (b) in a ring $R$ without identity is: $$J(R)=\{a\in R\mid ...
3
votes
1answer
4k views

proof of chinese remainder theorem for ring

Let $R$ be a ring (not necessary having "1"), and let $I,J$ be ideals of $R$ such that $I+J=R$. I want to prove that, for any $r, s \in R$, there is a $x\in R$ such that $$x\equiv r ({\rm mod} I) \...
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 ...
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 $1=i+j$, ...
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 ...
-1
votes
1answer
132 views
6
votes
2answers
2k views

Embedding of a ring into a ring with unity

I was reading the theorem on Embedding of a ring into a ring with unity which is as follows: Let R be ring and $R\times \mathbb Z=\{(r,n)|r\in R,n\in \mathbb Z\}$. This is a ring with addition ...
0
votes
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 ...
4
votes
2answers
190 views

Why is $S/Z$ a domain for the ideal $Z=\{z\in S\mid za=0,\;\forall a\in R\}$ in $S$?

Suppose $R$ is a rng with no zero divisors, not necessarily commutative. I know $R$ can be embedded into a ring $S:=\mathbb{Z}\times R$ by identifying $r\in R$ with $(0,r)\in S$. The operations on $S$ ...
13
votes
2answers
1k views

Existence of prime ideals in rings without identity

Let $R$ be a commutative ring (not necessarily containing $1$). Say that $R$ is the trivial ring if it has trivial (zero) multiplication. If $R$ is the trivial ring, then $R$ has no prime ideals (as ...
3
votes
2answers
304 views

An example of a ring without identity that does not contain any maximal ideal. [duplicate]

I'm trying to find an example of a ring without identity that does not contain any maximal ideal. Help me some hints.
6
votes
1answer
816 views

if $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring.

I'm having trouble with this homework problem (from Algebra by Hungerford). If $S$ is a ring (possibly without identity) with no proper left ideals, then either $S^2=0$ or $S$ is a division ring. ...
2
votes
2answers
130 views

For a commutative ring R without identity, there exists a∈R such that Ra≠R

Is this statement true? Then how to prove it? For a non trivial commutative ring $R$ without identity, there exists $a \in R\setminus \{0\}$ such that $Ra \not = R$
4
votes
1answer
601 views

Example of a finite ring with identity containing a ring without identity

What is an example of a finite ring $R$ with unity and a subring $S$ of $R$ that is not a ring with unity?
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 ...
3
votes
1answer
161 views

In a commutative ring without identity, is $(a)(b)\subset (ab)$ or $(ab)\subset (a)(b)$?

Let $R$ be a commutative ring without unity. Consider an ideal $(a)$ generated by $a\in R$. Note that $(a)=\{ra+na : r\in R, n\in \textbf Z\}$ since $R$ has no identity. I wonder if $(a)(b)\subset (ab)...
10
votes
5answers
2k views

Commutative rings without assuming identity

I was going through Exercises in Dummit&Foote, which does not assume identity in the definition of a ring, and reached the following exercise: Prove that in a Boolean ring ($a^2 = a$ for all $a$...
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 ...