# Questions tagged [rngs]

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

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### Looking for counterexamples (non-domains)

I stumbled upon this question Why doesn't $xa = x$ for all $x \in R$ imply that $a$ is the unit of $R$? and understood the given answers. But it got me thinking about a counterexample, and I was ...
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### Semiring of classes of associates of a ring/rng

Let's call elements that generate the same principal ideal of a ring/rng associates. $[a]$ is the equivalence class of associates of an element $a$ of a ring/rng. Let $A = \{[a],[b], ...\}$ be the ...
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### 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'...
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### 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)....
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### 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 ...
### 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 ...