Tagged Questions
3
votes
0answers
119 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 ...
4
votes
2answers
132 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 ...
3
votes
1answer
120 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 $S$ ...
3
votes
1answer
334 views
$I+J=R$, 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 ...
