# Questions tagged [rngs]

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

103 questions
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### Why are ideals more important than subrings?

I have read that subgroups, subrings, submodules, etc. are substructures. But if you look at the definition of the Noetherian rings and Noetherian modules, Noetherian rings are defined with ideals ...
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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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### Applications of rings without identity

Many courses and books assume that rings have an identity. They say there is not much loss in generality in doing so as rings studied usually have an identity or can be embedded in a ring with an ...
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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 unit. ...
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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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### 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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### 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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### 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 ...
3answers
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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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### 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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### 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 $\cdot$ ...
1answer
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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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### Non-unital rings: a few examples

Every ring I've ever heard of is unital, i. e., contains a (unique) element $a$ such that $xa = ax = x$ for every $x$ in it. However, some rings do not have such an element. What are they? P. S.: one ...
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### Commutative rings without assuming identity

I was going through Exercises in Dummit&Foote, which does not assume identity in the definition of a ring, and reached the following exercise: Prove that in a Boolean ring ($a^2 = a$ for all $a$...
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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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### 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 then ...
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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 ...
2answers
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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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### Embedding of a ring into a ring with unity

I was reading the theorem on Embedding of a ring into a ring with unity which is as follows: Let R be ring and $R\times \mathbb Z=\{(r,n)|r\in R,n\in \mathbb Z\}$. This is a ring with addition ...
1answer
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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 ...
1answer
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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. ...
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1answer
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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 ...
2answers
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### Every commutative ring of characteristic $p$ contains $\mathbb F_p$ as a subring?

I know that if a commutative ring with unity is of characteristic $p$ then it will contain $\mathbb F_p$ as a subring, but if the ring is commutative with characteristic $p$ and without unity then is ...
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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.
2answers
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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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### $R$ a ring with 1 and charac. $n>1$ (resp. 0) $\implies R$ contains a subring $\simeq \Bbb{Z}_n$ (resp. $\Bbb{Z}$). What happens if $R$ has no 1?!

Definition: Let $R$ be a ring. We say that $n\in \Bbb{Z}_+$ is the characteristic of $R$ if it is the least positive integer such that $n r=0$, for all $r\in R$ (here $nr$ denotes $r+r+\dots+r$, "$n$ ...
1answer
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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)...