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Questions tagged [rngs]

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

30
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5answers
2k views

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 ...
7
votes
1answer
715 views

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 show ...
26
votes
1answer
3k views

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 ...
26
votes
3answers
7k views

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. ...
10
votes
5answers
4k views

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 ...
6
votes
3answers
2k views

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 ...
8
votes
1answer
4k views

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 ...
7
votes
3answers
578 views

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 ...
3
votes
1answer
4k views

proof of chinese remainder theorem for ring

Let $R$ be a ring (not necessary having "1"), and let $I,J$ be ideals of $R$ such that $I+J=R$. I want to prove that, for any $r, s \in R$, there is a $x\in R$ such that $$x\equiv r ({\rm mod} I) \...
1
vote
2answers
250 views

Product of Principal Ideals when $R$ is commutative, but not necessarily unital

When $R$ is a ring (not necessarily commutative, and not necessarily with unity), I have a result that tells me that for $x \in R$, the ideal generated by $x$, $(x) $, is $= I_{x} = \langle RxR \...
19
votes
6answers
6k views

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. ...
8
votes
2answers
164 views

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 $A$?
13
votes
3answers
505 views

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 ...
3
votes
1answer
2k views

Non-commutative rings without identity [closed]

I'm looking for examples (if there are such) of non-commutative rings without multiplicative identity which have the following properties: 1) finite with zero divisors 2) infinite with zero divisors ...
0
votes
2answers
140 views

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 ...
2
votes
2answers
120 views

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 ...
17
votes
4answers
1k views

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 ...
6
votes
1answer
816 views

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. ...
6
votes
2answers
299 views

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 ...
8
votes
2answers
422 views

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 2\...
3
votes
3answers
171 views

$R$ 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 $r+r+\dots+r$, "$n$ ...
3
votes
1answer
85 views

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, $...
3
votes
1answer
106 views

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 $R'$?
1
vote
1answer
847 views

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 ...