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

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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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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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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 ...
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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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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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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 ...
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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 ...
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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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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 ...
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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 ...
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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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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, ...
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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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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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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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Equality in rng with no zero divisors.

I'm working on this problem, but I'm missing some manipulation. Suppose $R$ is a rng without zero divisors and has elements $a$ and $b\neq 0$ such that $ab+kb=0$ for some $k\in\mathbb{N}$ (that is, ...
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Theorem of Kaplansky, $R$ is a division ring if every element but one is (right) quasi-invertible.

There is a theorem of Kaplansky that seems to pop up every algebra book. Here rng denotes a ring with possibly no identity. As definition, an element $a$ of a rng $R$ is said to be (right) ...
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Does this proof that $2\mathbb{Z}\not\cong 3\mathbb{Z}$ work?

I'm trying to show $2\mathbb{Z}\not\cong 3\mathbb{Z}$ as rings. Suppose $\phi$ is an isomorphism. Then $\phi(2+2)=\phi(2\cdot 2)$, which implies $2\phi(2)=\phi(2)\phi(2)$. Then $2=\phi(2)$, which is ...
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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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Are there rngs whose rngs of matrices are commutative?

If $R$ is a unital ring and $M_{2\times 2}(R)$ is a commutative ring, then $R$ is a trivial ring because if $$\begin{pmatrix}0 & 1 \\ 0 & 0\end{pmatrix}=\begin{pmatrix}1 & 0 \\ 0 & ...
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Is there a domain without unity in which every element is a product?

In this answer, Fortuon Paendrag provides an example of a ring without unity such that every element is a product of some two elements. The example has zero divisors. Can a ring without a unity and ...
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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 ...
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$I+J=R$, 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 prove ...
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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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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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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 ...
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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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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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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 ...