Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [rngs]

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

The term rng (pronounced "rung") is used to refer to an associative ring which does not necessarily have an identity element. (rng = ring - i​dentity). Use this tag if your question particularly is about certain conditions forcing the existence of identity, or a morphism preserving identity. Using this tag along with or will be appreciated.

138 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 and Noetherian modules are defined with submodules.…
Gobi
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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. Nevertheless it is not prime because $2 \cdot 2 \in…
Joachim
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1 answer

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 identity. What then are the major applications of…
user4594
22
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6 answers

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. If it had said "finite commutative ring with…
Josh
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4 answers

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 unity to be a necessary part of ring, lets call that…
Bhaskar Vashishth
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6 answers

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 counterexamples, if any. The more examples, the…
Makoto Kato
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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 any ideal contains $0$, hence the square of every…
zcn
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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 recently came across the fact that every ring has to…
Jackson Kailath
  • 266
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3 answers

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 multiplicative unit element. I need a hint for this…
mathfan
  • 501
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If $I+J=R$, 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 that $IJ=I\cap J$. One inclusion is easy. If…
Daniel Montealegre
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7 answers

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 will notice I assumed commutativity. So, for an…
Luke
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2 answers

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 closest condition I can think of is the following: If…
user111064
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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 $\cdot$ distributes over $+$ in the usual way. Finally, for every…
Alexander Sibelius
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3 answers

In a ring, how do we prove that $a \cdot 0 = 0$?

In a ring, I was trying to prove that for all $a$, $a\cdot0 = 0$. 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 directly from the definition of a ring. Many thanks!
mareoraft
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1 answer

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. a module $P$ is projective if for any epimorphism $M\to N$ and any morphism $P\to N$ there exists a morphism $P\to M$ such that the…
Julian Kuelshammer
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