Prove That a Quotient of a Pid by a Prime Ideal Is Again a Pid

Algebraic structure

In mathematics, a master ideal domain, or PID, is an integral domain in which every platonic is principal, i.e., can be generated by a unmarried element. More generally, a principal platonic ring is a nonzero commutative ring whose ideals are chief, although some authors (east.thou., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal platonic band may take zero divisors whereas a primary ideal domain cannot.

Main platonic domains are thus mathematical objects that behave somewhat similar the integers, with respect to divisibility: whatever element of a PID has a unique decomposition into prime elements (then an analogue of the fundamental theorem of arithmetics holds); any two elements of a PID take a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If ten and y are elements of a PID without common divisors, then every element of the PID tin can be written in the form ax + by .

Principal ideal domains are noetherian, they are integrally airtight, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are master ideal domains.

Principal ideal domains announced in the following chain of class inclusions:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally airtight domains ⊃ GCD domains ⊃ unique factorization domains ⊃ master ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

Examples [edit]

Examples include:

Non-examples [edit]

Examples of integral domains that are not PIDs:

Modules [edit]

The central result is the construction theorem: If R is a master platonic domain, and M is a finitely generated R-module, then ${\displaystyle 1000}$ is a direct sum of cyclic modules, i.e., modules with one generator. The cyclic modules are isomorphic to ${\displaystyle R/xR}$ for some ${\displaystyle x\in R}$ ^[iv] (notice that ${\displaystyle ten}$ may be equal to ${\displaystyle 0}$ , in which instance ${\displaystyle R/xR}$ is ${\displaystyle R}$ ).

If Yard is a free module over a main ideal domain R, so every submodule of 1000 is once again costless. This does not agree for modules over capricious rings, as the instance ${\displaystyle (two,X)\subseteq \mathbb {Z} [X]}$ of modules over ${\displaystyle \mathbb {Z} [X]}$ shows.

Properties [edit]

In a principal ideal domain, any two elements a,b have a greatest mutual divisor, which may be obtained as a generator of the ideal (a, b).

All Euclidean domains are chief platonic domains, but the converse is not true. An example of a principal platonic domain that is not a Euclidean domain is the ring ${\displaystyle \mathbb {Z} \left[{\frac {one+{\sqrt {-xix}}}{2}}\right].}$ ^{[5] [half dozen]} In this domain no q and r exist, with 0 ≤ |r| < four, so that ${\displaystyle (i+{\sqrt {-xix}})=(4)q+r}$ , despite ${\displaystyle 1+{\sqrt {-xix}}}$ and ${\displaystyle four}$ having a greatest common divisor of 2.

Every principal ideal domain is a unique factorization domain (UFD).^{[7] [8] [ix] [10]} The converse does not concord since for whatever UFD K , the band K[X, Y] of polynomials in 2 variables is a UFD but is not a PID. (To bear witness this look at the ideal generated past ${\displaystyle \left\langle X,Y\right\rangle .}$ Information technology is non the whole band since it contains no polynomials of degree 0, but it cannot be generated by any one single element.)

1. Every principal ideal domain is Noetherian.
2. In all unital rings, maximal ideals are prime number. In principal ideal domains a near converse holds: every nonzero prime number ideal is maximal.
3. All principal platonic domains are integrally closed.

The previous three statements give the definition of a Dedekind domain, and hence every main platonic domain is a Dedekind domain.

Let A be an integral domain. And so the following are equivalent.

1. A is a PID.
2. Every prime ideal of A is principal.^[xi]
3. A is a Dedekind domain that is a UFD.
4. Every finitely generated ideal of A is principal (i.e., A is a Bézout domain) and A satisfies the ascending chain status on principal ethics.
5. A admits a Dedekind–Hasse norm.^[12]

Any Euclidean norm is a Dedekind-Hasse norm; thus, (5) shows that a Euclidean domain is a PID. (4) compares to:

• An integral domain is a UFD if and simply if it is a GCD domain (i.e., a domain where every two elements have a greatest common divisor) satisfying the ascending chain condition on principal ideals.

An integral domain is a Bézout domain if and only if whatever two elements in information technology have a gcd that is a linear combination of the two. A Bézout domain is thus a GCD domain, and (4) gives yet another proof that a PID is a UFD.

See too [edit]

• Bézout'due south identity

Notes [edit]

1. ^ Run into Fraleigh & Katz (1967), p. 73, Corollary of Theorem 1.7, and notes at p. 369, later on the corollary of Theorem 7.ii
2. ^ Run across Fraleigh & Katz (1967), p. 385, Theorem vii.8 and p. 377, Theorem 7.4.
3. ^ Milne. "Algebraic Number Theory" (PDF). p. 5.
4. ^ See too Ribenboim (2001), p. 113, proof of lemma 2.
5. ^ Wilson, Jack C. "A Principal Ring that is Not a Euclidean Ring." Math. Mag 46 (Jan 1973) 34-38 [1]
6. ^ George Bergman, A master ideal domain that is not Euclidean - adult as a series of exercises PostScript file
7. ^ Proof: every prime ideal is generated past 1 element, which is necessarily prime. Now refer to the fact that an integral domain is a UFD if and only if its prime number ideals contain prime number elements.
8. ^ Jacobson (2009), p. 148, Theorem two.23.
9. ^ Fraleigh & Katz (1967), p. 368, Theorem 7.2
10. ^ Hazewinkel, Gubareni & Kirichenko (2004), p.166, Theorem vii.2.1.
11. ^ T. Y. Lam and Manuel L. Reyes, A Prime number Ideal Principle in Commutative Algebra Archived 2010-07-26 at the Wayback Machine
12. ^ Hazewinkel, Gubareni & Kirichenko (2004), p.170, Suggestion 7.3.iii.

References [edit]

• Michiel Hazewinkel, Nadiya Gubareni, 5. 5. Kirichenko. Algebras, rings and modules. Kluwer Academic Publishers, 2004. ISBN 1-4020-2690-0
• John B. Fraleigh, Victor J. Katz. A first course in abstract algebra. Addison-Wesley Publishing Company. five ed., 1967. ISBN 0-201-53467-3
• Nathan Jacobson. Basic Algebra I. Dover, 2009. ISBN 978-0-486-47189-ane
• Paulo Ribenboim. Classical theory of algebraic numbers. Springer, 2001. ISBN 0-387-95070-2

External links [edit]

• Principal ring on MathWorld

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Source: https://en.wikipedia.org/wiki/Principal_ideal_domain

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