# Tagged Questions

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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### 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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### 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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### 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 ...
### 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. ...