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 called principal ideals.)
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.
Left Bézout rings are defined similarly.
A 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.
If R is a principal right ideal ring, then it is certainly a right Noetherian ring, since every right ideal is finitely generated.
It is also a right Bézout ring since all finitely generated right ideals are principal.
Principal right ideal rings are closed under finite direct products.
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.
This follows readily from the isomorphism theorems for rings.
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:
Then R is a finite local ring which is not principal.
forms a commutative principal ideal ring with unity, where
represents set symmetric difference and
If instead X is infinite, the ring is not principal: take the ideal generated by the finite subsets of X, for example.
Galois rings are commutative local PIRs.
in essentially the same way that finite field extensions of the integers modulo
The principal rings constructed in Example 5 above are always Artinian rings; 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.
The proof applies the Chinese Remainder theorem to a minimal primary decomposition of the zero ideal.
, 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.
Every semisimple ring R which is not just a product of fields is a noncommutative right and left principal ideal ring (it need not be a domain, as the example of n x n matrices over a field shows).
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 rings are seen to be both right and left Bézout rings.
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.