Intereting Posts

Another question on almost sure and convergence in probability
Riemann vs. Stieltjes Integral
Is There a Problem with This Professor's Proof Concerning Interior and Boundary Points?
Indefinite integral of $\log(\sin(x))$
Why is this congruence true?
Minimum value of $\cos x+\cos y+\cos(x-y)$
Solving 4th-order PDE
If a cyclic group has an element of infinite order, how many elements of finite order does it have?
Number of integral roots of a polynomial
computing the orbits for a group action
About finding $2\times 2$ matrices that are their own inverses
The diffential of commutator map in a Lie group
Inclusion of $l^p$ space for sequences
How do quaternions represent rotations?
How to prove a function is the Fourier transform of another $L^{1}$ function?

Possible Duplicate:

Proving that an ideal in a PID is maximal if and only if it is generated by an irreducible

I am trying to see whether the ideal generated by irreducible element in a principal ideal domain (PID) is maximal ideal.

Suppose ** r** is irreducible in a PID say

- let $H\subset G$ with $|G:H|=n$ then $\exists~K\leq H$ with $K\unlhd G$ such that $|G:K|\leq n!$ (Dummit Fooote 4.2.8)
- Determine the degree of the splitting field for $f(x)=x^{15}-1$.
- Prove that there is no element of order $8$ in $SL(2,3)$
- Group of order $p^2$ is commutative with prime $p$
- Maximal ideals of some localization of a commutative ring
- $|a|=m,\,\gcd(m,n)=1 \implies a$ is an $n$'th power

Let ** I** be an ideal of

Since ** D** is a principal ideal domain, there exist

So, ** r**=

If ** s** is a unit then

If ** t** is a unit then (

I need a little help for this. Thanks

- Why don't we study algebraic objects with more than two operations?
- Is this an isomorphism possible?
- Prove $4+\sqrt{5}$ is prime in $\mathbb{Z}$
- Why does $(A/I)\otimes_R (B/J)\cong(A\otimes_R B)/(I\otimes_R 1+1\otimes_R J)$?
- How to compute the distance of an affine space from the origin
- Conjugate class in the dihedral group
- Ideal of $\mathbb{C}$ not generated by two elements
- Show there is a surjective homomorphism from $\mathbb{Z}\ast\mathbb{Z}$ onto $C_2\ast C_3$
- How to multiply and reduce ideals in quadratic number ring.
- What About The Converse of Lagrange's Theorem?

In fact, we can generalize a bit.

**Proposition:** If $R$ is an integral domain and $x\in R\setminus \{0\}$, then $x$ is irreducible if and only if $xR$ is maximal amongst all principal proper ideals of $R$ (ie if $I=yR \subsetneq R$ and $xR\subseteq yR$ then $xR=yR$).

**Proof:** ($\Rightarrow$) Suppose $x$ is irreducible and choose $y\in R$ with $xR\subseteq yR\subsetneq R$. Then, for some $r\in R$, $x=yr$. Since $x$ is irreducible, either $y\in U(R)$ or $r\in U(R)$. However, the fact that $yR\neq R$ implies that $y\notin U(R)$, hence $r\in U(R)$, $y=r^{-1}x$, and it easily follows that $xR=yR$. Hence, $xR$ is maximal amongst proper principal ideals of $R$.

($\Leftarrow$) Suppose that $xR$ is maximal amongst all principal proper ideals of $R$, and assume that $x=yz$ for nonzero nonunits $y,z\in R$. Then, since neither $y$ nor $z$ are units, it is clear that $xR=yzR\subsetneq yR\subsetneq R$. This contradicts the maximality of $xR$ amongst proper principal ideals of $R$. Therefore $x$ is irreducible. $\blacksquare$

Now, if $R$ is a PID, then the above proposition implies that for all irredicuble $x\in R$, $xR$ is a maximal ideal.

In a PID, to divide is to contain: $b \mid a$ iff $(a)\subseteq(b)$. Thus, $(a)$ is maximal iff $a$ has no non-trivial divisors iff $a$ is irreducible.

An element is irreducible iff the ideal it generates is maximal amongst the principal ideals.

If all ideals are principal, then an element is irreducible iff the ideal it generates is maximal.

**HINT** $\ $ For principal ideals: contains $\iff$ divides. Hence, having no proper containing ideal (maximal) is equivalent to having no proper divisor (irreducible).

More generally, in domains where ideals satisfy contains $\iff$ divides (e.g. Dedekind domains), prime ideals $\ne 0$ are maximal. This characterizes PIDs, i.e. PIDs are precisely the UFDs where every prime ideal $\ne 0$ is maximal (i.e. Krull dimension $\le 1$).

- Does every infinite field contain a countably infinite subfield?
- The Area of an Irregular Hexagon
- Polynomials in one variable with infinitely many roots.
- Arithmetic mean of positive integers less than an integer $N$ and co-prime with $N$.
- Uniqueness for Integral Transform
- What does the notation $[0,1)$ mean?
- Prove that if a particular function is measurable, then its image is a rect line
- Monos in $\mathsf{Mon}$ are injective homomorphisms.
- Is the complement of countably many disjoint closed disks path connected?
- how to show that this complex series converge?
- Asymptotic expansion of the integral $\int_0^\infty e^{-xt} \ln(1+\sqrt{t}) dt$ for $x \to \infty$
- Sum over the powers of the roots of unity $\sum \omega_j^k$
- Textbook for Projective Geometry
- Principal maximal ideals in coordinate ring of an elliptic curve
- Understanding the differential $dx$ when doing $u$-substitution