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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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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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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? ...
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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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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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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 $A\subseteq P$ ...
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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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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 ...
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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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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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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 ...