1
vote
0answers
44 views

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?
8
votes
1answer
228 views

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 ...
3
votes
1answer
115 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 ...
3
votes
2answers
132 views

$r$ 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.
4
votes
1answer
51 views

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 ...
7
votes
3answers
323 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 ...
4
votes
0answers
101 views

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 ...
3
votes
3answers
104 views

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. ...
2
votes
2answers
274 views

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 ...
5
votes
1answer
85 views

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 ...
8
votes
6answers
358 views

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 ...
3
votes
0answers
166 views

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 ...
5
votes
2answers
152 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 ...
10
votes
3answers
271 views

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 $\cdot$ ...
3
votes
1answer
142 views

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 $S$ ...
2
votes
1answer
64 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
181 views

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) ...
3
votes
1answer
104 views

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 ...
1
vote
1answer
65 views

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 & ...
2
votes
2answers
170 views

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 ...
4
votes
1answer
322 views

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 ...
3
votes
0answers
164 views

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 ...
6
votes
1answer
307 views

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 ...
8
votes
2answers
384 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 ...
5
votes
1answer
654 views

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 ...
10
votes
3answers
395 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 ...
11
votes
5answers
2k 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: In a finite commutative ring every non-zero-divisor is a unit. If it ...
7
votes
1answer
559 views

Ring homomorphisms which map a unit to a unit map unity to unity?

this is the third part of a question I've been working on from Hungerford's Algebra. It is exercise 15 in the first section of Chapter III. $(c)$ If $f\colon R\to S$ is a homomorphism of rings ...
8
votes
5answers
1k 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 ...
5
votes
5answers
836 views

Finite rings of prime order must have a multiplicative identity

The standard definition of a ring is an abelian group that is a monoid under multiplication (with distributivity). However there are some books that have a weaker definition in that a ring only has to ...