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.

126 questions
10
votes
5 answers

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 $a$) every prime ideal is a maximal ideal. This was…
Carl
  • 3,408
  • 1
  • 18
  • 34
10
votes
3 answers

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 directly from the definition of a ring. Many thanks!
mareoraft
  • 373
  • 1
  • 5
  • 14
9
votes
3 answers

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 case this is not true in general, what is a…
kjo
  • 13,386
  • 9
  • 41
  • 79
9
votes
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. $P$ is projective if for any epimorphism $M\to N$ and any morphism $P\to N$ there exists a morphism $M\to P$ such that the diagram…
Julian Kuelshammer
  • 9,352
  • 4
  • 35
  • 78
8
votes
2 answers

Books on Rings without Identity

I was just wondering if anybody knows of any good books or articles that study rings (and algebras) without (or not necessarily with) identity. I have gone through Thomas Hungerford's Algebra textbook (and loved it), but every book I have read…
8
votes
1 answer

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 with identity and $u$ is a unit in $R$ such that $f(u)$…
yunone
  • 21,565
  • 7
  • 72
  • 153
8
votes
2 answers

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 $e-a \neq 0$. But $a(e-a) = 0$ so $a$ is zero…
Igor
  • 2,043
  • 14
  • 19
8
votes
2 answers

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. We've just proven in the previous part that if $S$…
smackcrane
  • 3,061
  • 2
  • 20
  • 32
8
votes
2 answers

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 $A$?
Rubertos
  • 11,841
  • 2
  • 19
  • 60
8
votes
2 answers

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
user23211
8
votes
2 answers

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 $2\in 2\mathbb Z$ isn't a product. It seems to be a major…
user23211
7
votes
5 answers

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 implying that a ring only has to be closed under multiplication (no…
crasic
  • 4,541
  • 6
  • 29
  • 29
7
votes
1 answer

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 $B\subseteq P$ (ii) $aRb\subseteq P$ with $a,b\in R$…
hmmmm
  • 5,612
  • 2
  • 38
  • 75
7
votes
1 answer

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 that $1$ and $0$ are the only idempotents in a ring…
Improve
  • 1,688
  • 2
  • 13
  • 24
7
votes
2 answers

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 $\mathscr{A} = \bigcap\{N\subset M: N \text{ is a submodule of } M , S\subset N\}$ Define $\mathscr{B} =…
Jj-
  • 1,696
  • 13
  • 24
1
2
3
8 9