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

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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 ...
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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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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 $...
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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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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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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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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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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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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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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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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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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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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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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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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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How can the rng $(\mathbb{N}, +, \cdot)$ be an ideal of some ring?

It is known than rng is an ideal of some ring. But how can rng $(\mathbb{N}, +, \cdot)$ be an ideal to some ring? As ring has an inverse element, the first ring we get from $\mathbb{N}$ is $\mathbb{Z}$...
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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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Example of a ring $R$ such that $R\otimes_R R\not\simeq R$

I want to show the following statement: If a ring $R$ is commutative and $I,J\triangleleft R$, then $$ R/I\otimes_R R/J\simeq R/(I+J). $$ I can easily show this, assuming that $1\in R$. What ...
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Is it possible to extend a commutative ring to have a unity? [duplicate]

Let $R$ be a commutative ring. Then, is it possible to extend this to have a unity? That is, is there a commutative ring with unity $R'$ such that $R$ is a subring of $R'$?
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$R$ 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 ideal?

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 ...
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Finitely generated idempotent ideal must be generated by an idempotent [duplicate]

Let $A$ be a commutative but not necessarily unital ring. How can we show that a finitely-generated ideal $I$ of a ring $A$ satisfying $I=I^2$ is generated by an idempotent element?
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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)...
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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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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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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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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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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 $Ra \not = R$
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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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$r$ 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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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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No nonzero proper ideals of $K$-algebra $A$ implies the ring $A$ has no nonzero proper ideals

This is from Seth Warner's Classical Modern Algebra. The problem is: If $A$ is a nontrivial $K$-algebra possessing no nonzero proper ideals, then there are no nonzero proper ideals of the ring $...
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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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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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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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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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Direct Product, and Subdirect Product in Herstein text

I'm reading Noncommutative Rings by I. N. Herstein. And I find one lemma pretty strange. It's on page 52 of the book. I'm typing everything necessary all here, so everyone can have a look at it. ...
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What does Herstein mean by 'centroid of a ring'?

I'm currently reading Herstein's Noncommutative Rings, and the definition of the centroid of a ring is on page 46 of the book. Let $\text{End}(R)$ be the ring of endomorphisms of the additive group $...
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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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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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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 ...
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Non-commutative rings without identity [on hold]

I'm looking for examples (if there are such) of non-commutative rings without multiplicative identity which have the following properties: 1) finite with zero divisors 2) infinite with zero divisors ...
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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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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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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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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. ...