Principal ideal ring
Encyclopedia
In mathematics
, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are principal ideal
s.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring
, R can be called a principal ideal ring, or simply principal ring.
If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring. Similarly left Bézout rings are defined. These conditions are studied in domains as Bézout domain
s.
A commutative principal ideal ring which is also an integral domain is said to be a principal ideal domain
(PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.
, since every right ideal is finitely generated. It is also a right Bézout ring since all finitely generated right ideals are principal. Indeed, it is clear that principal right ideal rings are exactly the rings which are both right Bézout and right Noetherian.
Principal right ideal rings are closed under finite direct product
s. If
, then each right ideal of R is of the form 
, where each 
is a right ideal of Ri. If all the Ri are principal right ideal rings, then Ai=xiRi, and then it can be seen that 
. Without much more effort, it can be shown that right Bézout rings are also closed under finite direct products.
Principal right ideal rings and right Bézout rings are also closed under quotients, that is, if I is a proper ideal of principal right ideal ring R, then the quotient ring R/I is also principal right ideal ring. This follows readily from the isomorphism theorem
s for rings.
All properties above have left analogues as well.
be rings and 
. Then R is a principal ring if and only if Ri is a principal ring for all i.
2. The localization of a principal ring at any multiplicative subset is again a principal ring. Similarly, any quotient of a principal ring is again a principal ring.
3. Let R be a Dedekind domain
and I be a nonzero ideal of R. Then the quotient R/I is a principal ring. Indeed, we may factor I as a product of prime
powers:
, and by the Chinese Remainder Theorem
, so it suffices to see that each
is a principal ring. But 
is isomorphic to the quotient 
of the discrete valuation ring
and, being a quotient of a principal ring, is itself a principal ring.
4. Let k be a finite field and put
, 
and 
. Then R is a finite local ring which is not principal.
s; in particular they are isomorphic to a finite direct product of principal Artinian local rings.
A local Artinian principal ring is called a special principal ring and has an extremely simple ideal structure: there are only finitely many ideals, each of which is a power of the maximal ideal. For this reason, special principal rings are examples of uniserial rings
.
The following result gives a complete classification of principal rings in terms of special principal rings and principal ideal domains.
Theorem (Zariski-Samuel): Let R be a principal ring. Then R can be written as a direct product
, where each Ri is either a principal ideal domain or a special principal ring.
The proof applies the Chinese Remainder theorem to a minimal primary decomposition of the zero ideal.
There is also the following result, due to Hungerford:
Theorem (Hungerford): Let R be a principal ring. Then R can be written as a direct product
, where each Ri is a quotient of a principal ideal domain.
The proof of Hungerford's theorem employs Cohen's structure theorems for complete local rings.
Arguing as in Example 3. above and using the Zariski-Samuel theorem, it is easy to check that Hungerford's theorem is equivalent to the statement that any special principal ring is the quotient of a discrete valuation ring.
s are seen to be both right and left Bézout rings.
If D is a division ring
and
is a ring endomorphism which is not an automorphism
, then the skew polynomial ring
is known to be a principal left ideal domain which is not right Noetherian, and hence it cannot be a principal right ideal ring. This shows that even for domains principal left and principal right ideal rings are different.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are principal ideal
Principal ideal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
s.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
, R can be called a principal ideal ring, or simply principal ring.
If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring. Similarly left Bézout rings are defined. These conditions are studied in domains as Bézout domain
Bézout domain
In mathematics, a Bézout domain is an integral domain in which the sum of two principal ideals is again a principal ideal. This means that for every pair of elements a Bézout identity holds, and that every finitely generated ideal is principal...
s.
A commutative principal ideal ring which is also an integral domain is said to be a principal ideal domain
Principal ideal domain
In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
(PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.
General properties
If R is a right principal ideal ring, then it is certainly a right Noetherian ringNoetherian ring
In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
, since every right ideal is finitely generated. It is also a right Bézout ring since all finitely generated right ideals are principal. Indeed, it is clear that principal right ideal rings are exactly the rings which are both right Bézout and right Noetherian.
Principal right ideal rings are closed under finite direct product
Direct product
In mathematics, one can often define a direct product of objectsalready known, giving a new one. This is generally the Cartesian product of the underlying sets, together with a suitably defined structure on the product set....
s. If
, then each right ideal of R is of the form , where each is a right ideal of Ri. If all the Ri are principal right ideal rings, then Ai=xiRi, and then it can be seen that . Without much more effort, it can be shown that right Bézout rings are also closed under finite direct products.Principal right ideal rings and right Bézout rings are also closed under quotients, that is, if I is a proper ideal of principal right ideal ring R, then the quotient ring R/I is also principal right ideal ring. This follows readily from the isomorphism theorem
Isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...
s for rings.
All properties above have left analogues as well.
Commutative examples
1. Let be rings and . Then R is a principal ring if and only if Ri is a principal ring for all i.2. The localization of a principal ring at any multiplicative subset is again a principal ring. Similarly, any quotient of a principal ring is again a principal ring.
3. Let R be a Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
and I be a nonzero ideal of R. Then the quotient R/I is a principal ring. Indeed, we may factor I as a product of prime
powers:
, and by the Chinese Remainder TheoremChinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
, so it suffices to see that each is a principal ring. But is isomorphic to the quotient of the discrete valuation ringDiscrete valuation ring
In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
and, being a quotient of a principal ring, is itself a principal ring.4. Let k be a finite field and put
, and . Then R is a finite local ring which is not principal.Structure theory for commutative PIR's
The principal rings constructed in Example 3. above are always Artinian ringArtinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
s; in particular they are isomorphic to a finite direct product of principal Artinian local rings.
A local Artinian principal ring is called a special principal ring and has an extremely simple ideal structure: there are only finitely many ideals, each of which is a power of the maximal ideal. For this reason, special principal rings are examples of uniserial rings
Serial module
In abstract algebra, a uniserial module M is a module over a ring R, whose submodules are totally ordered by inclusion. This means simply that for any two submodules N1 and N2 of M, either N_1\subseteq N_2 or N_2\subseteq N_1. A module is called a serial module if it is a direct sum of uniserial...
.
The following result gives a complete classification of principal rings in terms of special principal rings and principal ideal domains.
Theorem (Zariski-Samuel): Let R be a principal ring. Then R can be written as a direct product
, where each Ri is either a principal ideal domain or a special principal ring.The proof applies the Chinese Remainder theorem to a minimal primary decomposition of the zero ideal.
There is also the following result, due to Hungerford:
Theorem (Hungerford): Let R be a principal ring. Then R can be written as a direct product
, where each Ri is a quotient of a principal ideal domain.The proof of Hungerford's theorem employs Cohen's structure theorems for complete local rings.
Arguing as in Example 3. above and using the Zariski-Samuel theorem, it is easy to check that Hungerford's theorem is equivalent to the statement that any special principal ring is the quotient of a discrete valuation ring.
Noncommutative examples
Every semisimple ring R which is not just a product of fields is a noncommutative right and left principal ideal domain. Every right and left ideal is a direct summand of R, and so is of the form eR or Re where e is an idempotent of R. Paralleling this example, von Neumann regular ringVon Neumann regular ring
In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R withOne may think of x as a "weak inverse" of a...
s are seen to be both right and left Bézout rings.
If D is a division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
and
is a ring endomorphism which is not an automorphismAutomorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
, then the skew polynomial ring
is known to be a principal left ideal domain which is not right Noetherian, and hence it cannot be a principal right ideal ring. This shows that even for domains principal left and principal right ideal rings are different.