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

Algebraic structure

In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.eastward., can be generated past a unmarried element. More generally, a main platonic ring is a nonzero commutative band whose ideals are principal, although some authors (eastward.g., Bourbaki) refer to PIDs equally chief rings. The distinction is that a principal ideal ring may have zip divisors whereas a chief platonic domain cannot.

Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any chemical element of a PID has a unique decomposition into prime elements (so an counterpart of the primal theorem of arithmetic holds); any two elements of a PID accept a greatest common divisor (although information technology may non exist possible to notice it using the Euclidean algorithm). If x and y are elements of a PID without common divisors, and so every element of the PID can exist written in the form ax + past .

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

Principal ideal domains appear in the post-obit concatenation of class inclusions:

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

Examples [edit]

Examples include:

Non-examples [edit]

Examples of integral domains that are non PIDs:

Modules [edit]

The fundamental result is the structure theorem: If R is a principal ideal domain, and Thousand is a finitely generated R-module, so ${\displaystyle Thou}$ is a direct sum of circadian modules, i.eastward., modules with 1 generator. The circadian modules are isomorphic to ${\displaystyle R/xR}$ for some ${\displaystyle ten\in R}$ ^[4] (notice that ${\displaystyle 10}$ may exist equal to ${\displaystyle 0}$ , in which case ${\displaystyle R/xR}$ is ${\displaystyle R}$ ).

If M is a gratis module over a master platonic domain R, and then every submodule of Thousand is again free. This does not hold for modules over arbitrary rings, as the case ${\displaystyle (ii,X)\subseteq \mathbb {Z} [X]}$ of modules over ${\displaystyle \mathbb {Z} [10]}$ shows.

Properties [edit]

In a principal platonic domain, whatsoever ii elements a,b have a greatest mutual divisor, which may exist obtained every bit a generator of the platonic (a, b).

All Euclidean domains are principal ideal domains, only the converse is non truthful. An instance of a principal ideal domain that is not a Euclidean domain is the ring ${\displaystyle \mathbb {Z} \left[{\frac {i+{\sqrt {-xix}}}{two}}\right].}$ ^{[v] [six]} In this domain no q and r exist, with 0 ≤ |r| < 4, and then that ${\displaystyle (one+{\sqrt {-19}})=(4)q+r}$ , despite ${\displaystyle one+{\sqrt {-nineteen}}}$ and ${\displaystyle 4}$ having a greatest mutual divisor of 2.

Every main ideal domain is a unique factorization domain (UFD).^{[7] [eight] [9] [ten]} The converse does non concur since for any UFD K , the ring K[X, Y] of polynomials in two variables is a UFD but is not a PID. (To prove this look at the ideal generated by ${\displaystyle \left\langle X,Y\right\rangle .}$ It is not the whole band since it contains no polynomials of degree 0, but information technology cannot be generated by any one unmarried element.)

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

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

Permit A be an integral domain. Then the following are equivalent.

1. A is a PID.
2. Every prime number ideal of A is principal.^[11]
3. A is a Dedekind domain that is a UFD.
4. Every finitely generated ideal of A is principal (i.due east., A is a Bézout domain) and A satisfies the ascending chain status on principal ideals.
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 only if it is a GCD domain (i.eastward., a domain where every 2 elements have a greatest common divisor) satisfying the ascending chain status on principal ideals.

An integral domain is a Bézout domain if and only if any 2 elements in it have a gcd that is a linear combination of the two. A Bézout domain is thus a GCD domain, and (four) gives yet some other proof that a PID is a UFD.

See likewise [edit]

• Bézout's identity

Notes [edit]

1. ^ See Fraleigh & Katz (1967), p. 73, Corollary of Theorem 1.seven, and notes at p. 369, after the corollary of Theorem 7.2
2. ^ Meet Fraleigh & Katz (1967), p. 385, Theorem 7.8 and p. 377, Theorem vii.4.
3. ^ Milne. "Algebraic Number Theory" (PDF). p. 5.
4. ^ See also Ribenboim (2001), p. 113, proof of lemma ii.
5. ^ Wilson, Jack C. "A Master Ring that is Non a Euclidean Ring." Math. Mag 46 (Jan 1973) 34-38 [1]
6. ^ George Bergman, A primary ideal domain that is not Euclidean - developed as a series of exercises PostScript file
7. ^ Proof: every prime platonic is generated by one element, which is necessarily prime. Now refer to the fact that an integral domain is a UFD if and only if its prime ideals contain prime elements.
8. ^ Jacobson (2009), p. 148, Theorem 2.23.
9. ^ Fraleigh & Katz (1967), p. 368, Theorem 7.two
10. ^ Hazewinkel, Gubareni & Kirichenko (2004), p.166, Theorem vii.two.1.
11. ^ T. Y. Lam and Manuel Fifty. Reyes, A Prime number Ideal Principle in Commutative Algebra Archived 2010-07-26 at the Wayback Car
12. ^ Hazewinkel, Gubareni & Kirichenko (2004), p.170, Proposition 7.3.iii.

References [edit]

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

External links [edit]

• Main ring on MathWorld

franklinster1980.blogspot.com

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

Share this post