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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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Why are ideals more important than subrings?

I have read that subgroups, subrings, submodules, etc. are substructures. But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals ...
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3answers
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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. ...
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1answer
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Applications of rings without identity

Many courses and books assume that rings have an identity. They say there is not much loss in generality in doing so as rings studied usually have an identity or can be embedded in a ring with an ...
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Does a finite commutative ring necessarily have a unity?

Does a finite commutative ring necessarily have a unity? I ask because of the following theorem given in my lecture notes: Theorem. In a finite commutative ring every non-zero-divisor is a unit. ...
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4answers
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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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6answers
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Examples of a commutative ring without an identity in which a maximal ideal is not a prime ideal

In a commutative ring with an identity, every maximal ideal is a prime ideal. However, if a commutative ring does not have an identity, I'm not sure this is true. I would like to know the ...
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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 ...
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4answers
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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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3answers
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A finite commutative ring with the property that every element can be written as product of two elements is unital

I was struggling for days with this nice problem: Let $A$ be a finite commutative ring such that every element of $A$ can be written as product of two elements of $A$. Show that $A$ has a ...
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2answers
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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 ...
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3answers
380 views

Is there a name for this ring-like object?

Let $S$ be an abelian group under an operation denoted by $+$. Suppose further that $S$ is closed under a commutative, associative law of multiplication denoted by $\cdot$. Say that $\cdot$ ...
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1answer
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Pronunciation of `Rng` - the non-unital Ring

I chuckled the first time I heard that a Ring without a multiplicative identity (Ring without the i) is called a ...
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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 ...
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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 $a$...
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2answers
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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 ...
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2answers
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Idempotents in rings without unity

Suppose there are non-trivial idempotents in the ring without unity. Is it right that all of them are zero divisors? If we're given unitary ring with unity $e$ and $a$ is non-trivial idempotent then ...
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2answers
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What is an element of a rng called which is not the product of any elements?

Let $R$ be a non-unital ring. Let $F:R\times R\longrightarrow R$ be a function given by the formula $F(x,y)=xy.$ Let $r\not\in\operatorname{im}(F).$ Such elements can exists, for example $2\in 2\...
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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 $A$?
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1answer
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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 ...
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2answers
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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 ...
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1answer
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Ring homomorphisms which map a unit to a unit map unity to unity?

this is the third part of a question I've been working on from Hungerford's Algebra. It is exercise 15 in the first section of Chapter III. $(c)$ If $f\colon R\to S$ is a homomorphism of rings with ...
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The structure of a Noetherian ring in which every element is an idempotent.

Let $A$ be a ring which may not have a unity. Suppose every element $a$ of $A$ is an idempotent. i.e. $a^2 = a$. It is easily proved that $A$ is commutative. Suppose every ideal of $A$ is finitely ...
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1answer
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Idempotents in a ring without unity (rng) and no zero divisors.

Question: Given a ring without unity and with no zero-divisors, is it possible that there are idempotents other than zero? Def: $a$ is idempotent if $a^2 = a$. Originally the problem was to show ...
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5answers
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Finite rings of prime order must have a multiplicative identity

The standard definition of a ring is an abelian group that is a monoid under multiplication (with distributivity). However there are some books that have a weaker definition in that a ring only has to ...
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2answers
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How many commutative rings with exactly one non-zero zero divisor are there?

I recently rememebered the following theorem by Ganesan: Let $R$ be a commutative ring with $0<n<\infty$ non-zero zero divisors. Then $\operatorname{card}(R)\leq(n+1)^2.$ The proof proceeds ...
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1answer
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Equivalence of Definitions of Prime Ideal in Ring without $1$.

Let $R$ be a rng, so that $1\not\in R$. I am trying to show that following are equivalence of definition of prime ideal $P$; (i) $AB\subseteq P$ with $A,B\subseteq R$ implies $A\subseteq P$ or $B\...
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3answers
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Do Boolean rings always have a unit element?

Let $(B, +, \cdot)$ be a non-trivial ring with the property that every $x \in B$ satisfies $x \cdot x = x$. How does one prove that such a ring $(B, +, \cdot)$ must have a unit element $1_B$? (Or, in ...
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2answers
299 views

Is there any non-monoid ring which has no maximal ideal?

Is there any non-monoid ring which has no maximal ideal? We know that every commutative ring has at least one maximal ideals -from Advanced Algebra 1 when we are study Modules that makes it as a very ...
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2answers
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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 ...
6
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1answer
367 views

Prove that every commutative infinite rng $R$ has an infinite subrng $S$ s.t $S\neq R$

Prove that every infinite commutative rng $R$ has an infinite subrng $S$ such that $R\neq S$. (Where the rng is not defined to have the identity as a member). Any help or hints of how to go about ...
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1answer
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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. ...
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2answers
560 views

What is the definition of 'span' in a module?

$\newcommand{\supp}{\operatorname{supp}} \newcommand{\span}{\operatorname{span}}$Let $M$ be a module over a ring $R$ and $S\subset M$ Define $\mathscr{A} = \bigcap\{N\subset M: N \text{ is a ...
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1answer
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Simple + Artinian = Semiprimitive

By a noncommutative ring I mean that it has no unit. I know that if some ring (say, $R$) is simple, then: $R^2 \neq (0)$ It only possesses $2$ two-sided ideals, namely $(0)$, and itself. And that, ...
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1answer
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Projective modules over rings without unit

For rings with unit there are at least three ways to define a projective module: The universal property, i.e. $P$ is projective if for any epimorphism $M\to N$ and any morphism $P\to N$ there exists ...
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1answer
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Where in the proof did Herstein use the fact that $A$ is a two-sided ideal of $R$?

I'm reading Noncommutative Rings by I. N. Herstein. The theorem I'm having trouble with is 1.2.5, on page 16 of the book. Some definition 1. Regular ideal An ideal $\rho \subset R$ is called a ...
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3answers
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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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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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2answers
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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 ...
4
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1answer
280 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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1answer
600 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?
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2answers
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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 $S$ ...
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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 ...
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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
76 views

If $R$ is a rng, show that $R\times \mathbb{Z}$ contains a subset in one to one correspondence with $R$.

Let $(R,+,\cdot)$ be a rng (satisfies all the axioms of a ring except multiplicative identity). Define addition and multiplication in $R\times\mathbb{Z}$ by: $(a,n)+(b,m)=(a+b,n+m)$ and $(a,n)\cdot(b,...
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1answer
221 views

In a PID without unit an ideal is maximal iff it is prime.

As the titles says, I need to show that in a PID $R$ an ideal is maximal iff it is prime. This is easy to do if $R$ has a multiplicative identity. I can not do it if $R$ does not have an identity. It ...
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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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2answers
302 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.
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2answers
246 views

$r$ is not nilpotent, $r-r^2$ is nilpotent, then the ring has a non-zero idempotent

Assume that $R$ is a ring and $r-r^2$ is nilpotent for an element $r\in R$. If $r$ is not nilpotent, then $R$ has a nonzero idempotent.
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3answers
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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
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)...